LIBRARY OF CONGRESS. 



JO.. 



Shelf .._.S.4.4 

UNITED STATES OF AMERICA. 



ELEME]^TAET 



OO-OEDINATE GEOMETRY, 



COLLEGIATE USE AND PEIYATE STUDY. 



WILLIAM BENJAMIN SMITH, Ph.D. (Gottingen), 

PROFESSOR OF PHYSICS, MISSOURI STATE UNIVERSITY. 
MISSOURI. 



j b 



MAX/MUM REASONING. MINIMUM RECKONING. 



BOSTON: 
GINN & COMPANY. 

1886. 




Entered according to Act of Congress, in the year 1885, by 

WILLIAM BENJAMIN SMITH, 
in the Office of the Librarian of Congress, at Washington 



J. S. CtJsHiSTG & Co., Pbintbrs, Boston. 



PEEFACE. 



IN the study of Analytic Geometry, as of almost anything 
else, either or both of two ends may be had in view : 
gain of knowledge, culture of mind. While the first is in 
itself worthy enough, and for mathematical devotees all suffi- 
cient, it is certainl}' of only secondary importance to the 
mass of college students. For these the subject can be 
wisely prescribed in a curriculum only in case the mental 
drill it affords be very high in order of excellence. 

The worth of mere calculation as an exercise of reason 
can hardly be considerable, for reason is exercised onl}^ in 
a tread-mill fashion. Even the solution of problems b}^ al- 
gebraic processes is a very inferior discipline of reason, for 
only in forming the analytic statement does the reasoning 
rise clearly into consciousness ; the' operations that follow 
conduct one to the conclusion, but — with his eyes shut. In 
this respect Geometry is certainly a better discipline than 
Algebra, and the Euclidean than the Cartesian Geometry. 
But not in any kind of reasoning is the very best discipline 
found. No argument presents difficulty or calls for much 
mental effort to follow it, when once its terms are clearly 
understood ; for no such argument can be harder to under- 
stand than the general S3-llogism of which it is a special 
case, and that is of well-known simplicity. The real diffi- 
culty lies in forming clear notions of things ; in doing this 
all the higher faculties are brought into play. It is this 
formation of concepts, too, that is the really important part 
of mental training. He who forms them clearl}' and accu- 
rately may be safely trusted to put them together correctly. 



IV PKEFACE. 

Logical blunders are comparatively rare. Nearly every 
seeming mistake in reasoning is really a mistake in concep- 
tion. If this be false, that will be invalid. 

It is considerations like the above that have guided the 
composition of this book. Concepts have been introduced 
in abundance, and the proofs made to hinge directly upon 
them. Treated in this way, the subject seems adapted as 
hardly any other to develop the power of thought. 

The correlation of algebraic and geometric facts has been 
kept clearly and steadily in view. While each may be taken 
as pictures of the other, the former have generally been 
treated as originals, lending themselves much more readily 
to classification. 

Only natural logical order has been aimed at in the devel- 
opment of the subject ; no attempt has been made to keep 
up the distinctions of ancient and modern, analytic and 
s^mthetic. 

With every step forward in G-eometry the difficulty and 
tedium of graphical representation increases, while more and 
more the reasoning turns upon the form of the algebraic 
expressions. Accordingl}', pains have been taken to make 
the notation throughout consistent and suggestive, and Deter- 
minants have been used freely. 

By all this effort to make the book an instrument of 
culture, its worth as a repertory of mathematical facts has 
scarcely suffered ; in this regard, as in others, comparison 
with other texts is invited. 

AUTHOR. 

Central College, Mo. 
March 25, 1885. 



TABLE OF CONTENTS. 



INTRODUCTION. 

DETERMINANTS. 

Article. Page. 

1. Permutations, Straight and Circular, of n Things xiii 

. 2. Inversions in Permutations xiv 

3. Exchange of Elements in a Permutation xiv 

4. Definition of Determinant xv 

5. Ways of writing Determinants xvi 

6. Simplest Properties of Determinants xvii 

7. Co-factor of an Element of a Determinant xviii 

8. Products of Elements by their Co-factors xix 

9. Decomposition of a Determinant , xx 

10. Evaluation of a Determinant. Examples xxi 

11. Multiplication of Determinants xxiii 

12. Determinant of the Co-factors xxiv 

13. Solution of a System of Linear Equations xxv 

14. Conditions of Consistence of Equations xxvi 

15. Elimination between Equations of Higher Degree xxvii 



PART I. THE PLANE. 



CHAPTER I. 
FIRST NOTIONS. 

1. Definition of Functions 1 

2. Algebraic Expression of a Eunctional Relation. Argument 2 

3. Classification of Functions 3 

4. Definition of Continuity. Illustrations 4 

5. Numbers pictured by Tracts ; Addition and Subtraction 6 

6. Pairs of Numbers pictured by Points 7 

7. Equations pictured by Curves 8 



Vi TABLE OF CONTENTS. 

Article. Page. 

8. Determination of Position on a Surface 9 

9. Points pictured by Pairs of Numbers. Co-ordinates 10 

10. Polar Co-ordinates 11 

11. Polar Equations pictured by Curves 12 

12. Degrees of Freedom. The Picture changes with the System ... 12 

13. Summary 14 

14. Definition of Co-ordinate Geometry 14 

N.B. As to Real and Imaginary Numbers 15 



CHAPTER II. 

THE RIGHT LINE. 

15. Distance between Two Points 16 

16. Intersection of Curves 17 

17. Division of a Tract in any Ratio 18 

18. Parallel Projection of a Tract 20 

19. Projection of a Polygon 21 

20. Co-ordinates as Projections 22 

' 21. Formulae of Transformation 22 

22. Note on the Formulae 25 

23. Linear Substitution 25 

24. Equation of 1st Degree pictures a Right Line , . ... 28 

25. Special Forms of the Equation 29 

26. Angle between Two Right Lines 32 

27. Distance from a Point to a Right Line 34 

28. Families of Right Lines 35 

29. Pencils of Right Lines. 38 

30. The Equation fi^L^ -f /X2Z2 + /n-^L^ — O. Triangular Cds 39 

31. Constant Relation between Triangular Co-ordinates 40 

32. Geometric Meaning of the Parameter A 42 

33. Halvers of the Angles between Two Right Lines 43 

34. Illustrations of Abridged Notation 43 

35. Polar Equation of the Right Line 45 

36. Area of a Triangle given by its Vertices 46 

37. Area of a Polygon given by its Vertices 47 

38. Area of a Triangle given by its Sides 48 

39. Ratio in which a Tract is cut by a Right Line 48 

40. Theorems on Transversals - 49 

41. Cross Ratios of Ranges and Pencils 50 

42. Cross Ratios of Rays given by their Equations 54 

43. Equation of Condition that Four Rays be Harmonic 55 



TABLE OF CONTENTS. vii 

Article. Page. 

44. Relative Position of Pour Harmonics 55 

45. nomographic Pencils 56 

46. Common Harmonics 57 

47. Involution of Rays (Points) 57 

48. Centre and Foci of Involution 58 

49. Cross Ratio in Involutions 59 

50. Homogeneous Equation of Nth Degree 60 

51. Resolution of Equations of Higher Degree 60 

52. Angles between the Pair kx^ + 2hxy -{-jt/^ = 62 

53. Halvers of these Angles 63 

54. Intersection of Right Line and Curve of 2d Degree 63 

55. The Right Line as a Locus. Examples 64 

56. Families of Right Lines through a Point 69 



CHAPTER in. 

THE CIRCLE. 

57. Equation of the Circle 71 

58. Determination of a Circle 71 

59. Normal Form of the Equation of a Circle 72 

60. The Circle conditioned 73 

N.B. On the Quadratic Equation 74 

61. Axial Intercepts of the Circle 75 

62. Polar Equation of the Circle 76 

63. Intersection of Circle and Right Line 76 

64. Coincident Points 77 

65. Tangent to the Circle 79 

66. Tangent to Curve of 2d Degree 80 

67. Tangents from a Point to Curve of 2d Degree 81 

68. Outside and Inside of a Curve 82 

69. Polar and Pole. Conjugates 84 

70. Construction of Polar 85 

71. Polar as Locus of a 4th Harmonic 85 

72. Equation of a Pair of Tangents 87 

73. Diameter and Normal 88 

74. Power of a Point as to a Circle 89 

75. Relative Position of Pole and Polar 89 

76. Power-Line 92 

77. Power-Centre 93 

78. Power-Line of a System 94 

79. Equation of a System 94 



Viii TABLE OF CONTENTS. 

Article. Page. 

80. Limiting Points 94 

81. Orthogonal Circles 95 

82. Similar Figures 99 

83. Circles are Similar 100 

84. Chords of Contact 101 

85. Axes of Similitude 101 

86. Centrode of a Circle cutting 3 Circles under Y a 102 

87. The Taction-Problem proposed 103 

88. The Taction-Problem solved 104 

89. The Circle as Locus. Examples , 106 



CHAPTER IV. 
GENERAL PROPERTIES OP CONICS. 

90. Results recalled 113 

91. Distance from a Point to a Conic 114 

92. Diameter and Centre 114 

93. Centre and Diameter 115 

94. Direction of Diameters 116 

95. Central Equation of the Conic 117 

96. Axes of the Conic 117 

97. Criterion of the Conic 118 

98. Asymptotes of the Conic 119 

99. Central Distances of Pole and Polar 119 

100. Equation of P reduced 120 

101. Ratio of Distance Products 120 

102 Constant Functions of k, h,j, ca 123 



CHAPTER Y. 
SPECIAL PROPERTIES OP CONICS. 

103. Equations of Centrics in Intercept Form 125 

104. Relation of a Centric to its Axes 126 

105. Central Polar Equations of Centrics 127 

106. The Centrics traced 127 

107. The Equations solved as to y 129 

108. Direction of Conjugate Diameters 130 

109. Co-ordinates of Ends of Conjugate Diameters 131 

110. Squared Half-Diameters 132 

111. Central Distance of Tangent 132 



TABLE OF CONTENTS. ix 

Article. Page. 

112. Tangents, Subtangents, Normals, Subnormals 134 

113. Perpoles and Perpolars 136 

114. Foci and Directrices 137 

115. Polar and Perpolar as Bisectors 138 

116. Focal Radii 138 

117. Asymptotic Properties 141 

118. Asymptotic Equation oi H 142 

119. Polar Equations of £ and // 143 

120. P as the Limit of £ and // 145 

121. Properties of P as such Limit 146 

122. Tangent and Normal to Z' 148 

123. Interpretation of 45-' 149 



CHAPTER VI. 
SPECIAL METHODS AND PROBLEMS. 

124. Magic Equation of Tangent 151 

125. Magic Equation of Normal 152 

126. Illustration of Use of Magic Equations 153 

127. Eccentric Equation of £ 153 

128. Eccentric Equation of Chord, Tangent, Normal 154 

129. Quasi-Eccentric Equation oi H 155 

130. Hyperbolic Functions 157 

131. Supplemental Chords 158 

132. Auxiliary Circles of £ and H 158 

133. Their Correspondents in P 159 

134. Vertical Equation of the Conic 160 

135. Tangent Lengths from a Point to 2l P 161 

136. Areas in the £ 162 

137. Areas in the P 162 

138. Areas in the H 163 

139. Varieties of Conies 166 



CHAPTER VII. 

SPECIAL METHODS AND PROBLEMS. — ( Con^wwed.) 

140. Conditions fixing a Conic 168 

141. Conic through 5 Points 169 

142. Conies through 4 Points 170 

143. Pascal's Theorem 171 



X TABLE OF CONTENTS. 

Article. Page, 

144. Construction of the Conic by Pascal's Theorem 173 

145. Elements of the Centric 173 

146. Elements of the Non-Centric 176 

147. Construction of Conies 177 

148. Conf ocal Conies 179 

149. Conf ocals as Co-ordinate Lines 180 

150. Similar Conies 181 

151. Central Projection 183 

152. Projection of Y s into Y s given in Size 184 

153. The Conic a Central Projection of a Circle 185 



CHAPTER YIII. 
THE CONIC AS ENVELOPE. 

154. Homogeneous Co-ordinates 187 

155. Line-Co-ordinates 188 

156. Equations between Line-Co-ordinates 189 

157. Interchange of Equations 190 

158. Tangential Equation of 2d Degree 191 

159. Double Interpretation 193 

160. Brianchon's Theorem '. 193 

161. Loci of Poles and Envelopes of Polars 195 

Note on Points and Lines at co 196 

Examples 197 



PAET II. OF SPACE. 



CHAPTER I. 

1. Triplanar Co-ordinates 223 

2. Cylindric Co-ordinates 224 

3. Spheric Co-ordinates 225 

4. Direction-Cosines 226 

4* Projections with Oblique Axes 228 

5. Division of a Tract 231 

6. Transformation of Co-ordinates 231 

7. General Theorems 233 

8. Equation of the Right Line 234 



TABLE OP CONTENTS. xi 

Article. Page. 

9. Intersecting Right Lines 235 

10. Common Perpendicular to Two Right Lines 237 

11. Co-ordinates of a Right Line 238 

12. Equation of the Plane 238 

13. Normal Form of the Equation 239 

13* The Normal Eorm with Oblique Co-ordinates 239 

14. The Triangle in Space 240 

15. Position-Cosines 241 

15* Position-Cosines with Oblique Co-ordinates 241 

16. Intersecting Planes 241 

17. Lines on a Surface 243 

18. Three Planes 243 

19. Pour Planes 243 

20. Meaning of \ 244 

21. Right Lines halving V between Two Right Lines 244 

22. Pencils and Clusters of Planes 245 

23. Distance between Two Right Lines 246 

24. Tetraeder fixed by Planes 247 

25. Meaning of Co-ordinates of a Right Line 247 

CHAPTER II. 

26. Generation of Surfaces 249 

27. Cylindric Surfaces 25.0 

28. Conic Surfaces 250 

29. Surfaces of Revolution 251 

30. Discriminant of the Quadric 251 

31. Centre of the Quadric 252 

32. Tangents to the Quadric 253 

33. Tangent Cones 254 

34. Poles and Polars 254 

35. Diameters 255 

36. The Right Line and the Quadric 255 

37. Perpendicular Conjugates 256 

38. Rectangular Chief Planes 258 

39. Special Case 259 

40. Classification of Quadrics 260 

41. Ratios of Distances to the Quadric 261 

42. Invariants of the Quadric 261 

43. Geometric Interpretation 262 

44. The Ellipsoid 263 

45. Cyclic Planes 264 



Xll TABLE or CONTENTS. 

Article. Page. 

46. The Ellipsoid a Strained Sphere 265 

47. Eccentric Equation of the Tangent Plane 266 

48. Normal Equation of the Tangent Plane , 266 

49. The Simple Hyperboloid 267 

50. The Double Hyperboloid 267 

51. The Hyperboloids as Strained Equiaxials 268 

52. The Elliptic Paraboloid ., 268 

53. The Hyperbolic Paraboloid 269 

54. Imaginary Cyclic Planes 269 

55. The Quadric as ruled 270 

56. Right Lines on the Simple Hyperboloid 271 

57. Imaginary Right Lines on the Ellipsoid 272 

58. Right Lines on the Hyperbolic Paraboloid 272 

59. Eoci and Conf ocals 273 

60. Confocal Co-ordinates 274 

61. Cubature of the Hyperboloids 274 

62. Cubature of the Ellipsoid 276 

63. Cubature of the Elliptic Paraboloid 276 

64. Cubature of the Hyperbolic Paraboloid 277 

65. Determination of the Quadric 278 



INTRODUCTION. 

DETERMINANTS. 

Permutations. 

1. Two things, as a and 5, may be arranged straight, in the 
order of before and after, in but tivo ways : a b, b a. A tJiird 
thing, as c, may be introduced into each of these arrangements 
in tJiree ways : just before each or after all. Like may be said 
of any arrangement of n things; an (?i4-l)th thing maybe 
introduced in n -{-1 ways, namely, before each or after all. 
Hence the number of arrangements of n + 1 things is n -{-1 times 
the number of arrangements of n things. Or 



Writhig n ! for the product of the natural numbers up to w, 
we have 

71 -f- 1 ! = ?i ! n + 1. 



Hence, if P^ = n\, P^^i = n + 1 !, and so on. 
Now Ps = 2 = 1 . 2 = 2 ! ; 

hence, P3=l •2-3 = 3!, and Pn = nl 

The various arrangements of things in the order of before 
and after are called straight Permutations or simply Permuta- 
tions of the things. The number of permutations of n things is 
n ! (read factorial n or n factorial) . 

If the things be arranged not straight but around, in a ring, 
we ma}' suppose them strung on a string ; if there are n of 
them, there are also n spaces between them. We may suppose 
the string cut at any one of the n spaces and then stretched 
straight ; this will turn the circular permutation into a straight 
one ; and since we may make n different cuts, each yielding a 



XIV CO-ORDINATE GEOMETRY. 

distinct straight permutation, the number of straight permuta- 
tions of n things is ii times the number of circular ones. Or 



Pn = G^'n] .-. C'„ = 91 — 1 ! 

2. The things, whatever they be, are most conveniently 
marked or named by letters or numbers. Of letters the alpha- 
betic order is the natural order ; of numbers the order of size is 
the natural order ; as : a, 5, c, ••• ;2; ; 1, 2, 3, 4, ••• 7^. 

If any change be made in either of these orders, say in the 
last, then some less number must appear after some greater, 
some greater before some less. Every such change from the 
natural order is called an Inversion. The number of inversions 
in any permutation is found by counting the number of numbers 
less than a number and placed after it, and taking the sum of 
the numbers so counted. 

A permutation is'named even or odd^ according as the number 
of inversions in it is even or odd. Thus 2 5 3»1 6 4 is an even 
permutation containing 6 inversions ; 3 1 2 5 4 6 is an odd 
permutation containing 3 inversions. The natural order, 
1, 2, 3, ••• n, contains inversions and is even; the counter 

order, ?z ••• 3, 2, 1 contains l4-2 + 3-|-----J-7i — 1 or — '- 

1-2 

inversions and is even when the remainder on division of n by 4 
is or 1, odd when the remainder is 2 or 3. 

It is plain that in any permutation any thing, symbol, or ele- 
ment ma}^ be brought to any place or next to any other one by 
exchanging it in turn with each of the ones between it and that 
other one. Thus, in 374516 2, 7 may be brought next to 6 
by exchanging it in turn with 4, 5, 1. Hence any permutation 
may be produced from anj' other by exchanges of adjacents. 

3. By an exchange of any two adjacents, as j9, g, the rela- 
tions of each to all the others, and the relations of all the others 
am.ong themselves, are not changed ; onh' the relation of those 
two is changed. Novy if pg be an inversion, qj^ is not; and if 
pq be not an inversion, qp is one ; hence in either case, by this 



INTRODUCTION. XV 

exchange of two acljaceuts, the number of inversions is changed 
hy 1 ; hence the permutation changes from even to odd or from 
odd to even. 

li p and q be ?io?i-adjacent, and there be h elements between 
them, then p is brought next to g by A; exchanges in turn with 
adjacents, and then q is brouglit to ^'s former place by fc-j-l 
exchanges with adjacents : p is carried over k elements and q 
over A: + 1 ; thus p and q are made to exchange places by 2 Z: + 1 
exchanges of adjacents. The permutation meanwhile changes, 
from even to odd or from odd to even, 2A; + 1 times; and an 
odd number of changes back and forth leaves it changed. Hence, 
an exchange of any tico elements in a permutation changes the 
permutation from even to odd or from odd to even. 

Plainly all the permutations may be parted into pairs, the mem- 
bers of each pair being alike except as to p and q, which are 
exchanged in each pair ; hence one permutation of each pair 
will be even, and one odd; hence, of all the permutations, half 
are even, 7ia// odd. 

Determinants. 

4. It is plain that 9'r things ma}' be parted into n classes of 7i 
each. We maj mark these classes by letters : a, h^c, ••• ?i, where 
it is understood that n is the ?ith letter ; the rank of each in its 
class may be denoted by a subscript ; thus jh will be the A;th 
member of class j9. Clearly all members of rank k will also 
form a class of n members. The whole number may be thought 
arranged in a square of n rows and n columns, as in the special 
case 71 = 5, thus : 



«1 


h 


Ci 


ch 


ei 


ao 


6o 


Co 


do, 


e.-y 


ttg 


^3 


Cs 


cZ, 


63 


a^ 


h 


Ci 


d. 


^4 


a, 


h 


C-o 


d, 


e,5 



This arrangement is not at all necessary to our reasoning, but 
is quite convenient. 



XVI CO-ORDINATE GEOMETRY. 

Suppose we pick out of these n^ things n of them, taking one 
of each class and one of each rank (clearly, then, we take only 
one of each) . This we may do in n ! ways ; for we may write 
off the n letters in natural order, a, 6, c, ••• n, and then suffix 
the subscripts in as many ways as we can permute them, i.e., 
in n ! ways. 

Now suppose these iv' things symbolized b}^ letters to be mag- 
nitudes or numbers, and form the continued product of each set 
of n picked out as above : write off the sum of these products, 
giving each the sign + or — according as the permutation of 
the subscripts be even or odd : the result is called the Deter- 
minant of the magnitudes so classified. Accordingly a Deter- 
minant may be defined as 

A sum of products of n^ symbols assorted into n classes of 
n ranks each,, formed of factors taken one from each class and 
each rank^ each product marked -{- or — according as the order of 
ranks (or classes) is an even or an odd permutation^ the order 
of classes (or ranks) being natural. 

The symbols are called elements of the Determinant ; each 
product, a term; the number of the degree of the Determinant is 
the number of factors in each product. The classes may be 
denoted by letters and the ranks by subscripts, or vice versa. 
The definition shows that classes and ranks stand on exactly 
like footing ; in any reasoning they may be exchanged. 

5. There are several ways of writing Determinants. In the 
square way, exemplified in Art. 4, the classes are written, in 
columns and the ranks in rows., or vice versa. Hence roivs and 
columns are always interchangeable. This is a very vivid way 
of writing them, but is tedious. It is shorter to write simply 
the diagonal term, thus : 

%±a^b2C^"'n^ 

The sign of summation % refers to the different terms got by 
permuting the subscripts^ — there are n ! of them ; the double 



INTRODUCTION. Xvil 

sign ± means that each term is to be taken + or — according 
as the permutation of the subscripts is even or odd. 

Still another way is to write the diagonal between bars : 

This is very convenient when there can be no doubt as to 
what are the elements not written : otherwise, the square form 
is best. 

6. To exchange two rows in the square form would clearly 
be the. same as to exchange in ever}' term the indices or sub- 
scripts that mark those rows ; but by Art. 3 this would change 
each permutation of the subscripts from even to odd or from 
odd to even ; and this, b}' the definition, would change the sign 
of each term, and hence of the whole Determinant. Moreover, 
since rows and columns stand on like footing, the same holds 
of exchanging two columns : hence, 

To exchange two columns (or rows) changes the sign of the 
Determinant. 

If the two rows (or columns) exchanged be identical or con- 
gruent, i.e., if the elements corresponding in position in the 
two be equal each to each, clearly exchanging them can have 
no effect on the value of the Determinant, although it changes 
the sign ; now the only number whose value is not changed b}^ 
changing its sign is : hence, 

The value of a Determinant with tioo congruent roivs (or col- 
umns) is 0. 

Every term of a Determinant contains one and only one 
element out of each row and column ; hence a common factor 
in every element of a row (or column) must appear as a factor 
of every term of the Determinant and hence of the Determinant 
itself ; hence, we may divide each element of the row (or col- 
umn) by it, if at the same time we multipl}' the whole Determi- 
nant b}' it; i.e., cm?/ factor of every element of a row (or col- 



xvm co-ohdinatb geometry. 

umn) of a Determinant may he set out aside as a factor of the 
whole Determinant, 

It is equally plain that any factor may be introduced into each 
element of a row (or column) , if at the same time the whole 
Determinant be divided by that factor. 

7. If we will find all the terms that contain any one element of 
the Determinant, say a^, we may suppose all the other elements 
in its column and row to be ; this will make vanish no term 
containing % and all terms not containing a^. The Determi- 
nant, say of otli degree, will then be 





cti 





ho C2 


ck 


hi c. 


d, 



64 C4 d^ 64 
65 C5 d^ e^ 

Setting aside cz-i as the first factor in each product, we find all 
the part-products by holding the order hcde fast and permuting 
the subscripts 2 3 4 5 5 ^^^^ this is the wa}' we form the Deter- 
minant I&2C3 (^465! ; also the sign of each whole product, after 
multiplying by a^, will be the same as the sign of the corre- 
sponding part-product, since 1 being in its natural place, the 
only possible inversion will be in the subscripts 2345- Hence, 
the sum of the part-products or multipliers of % is the Deter- 
minant I &2C3(^4e5 1. It is called the co-factor (or sub-determinant, 
or minor) of % and is the Determinant left after destroying the 
row and column of %. 

If now we will find the co-factor of pj,, i.e., of the element in 
the j9th column and kth row, we may bring its row to the first 
place by exchanging its row in turn with each row before it, and 
its column in turn with each column before it. By these 
2^ -\-Jc— 2 exchanges the positions of the other rows and col- 
umns as to each other are not changed at all ; they stand exactly 
as if the j^^th column and ytth row had been destroyed. The new 
Determinant got by these exchanges will be the old one term 
for term with like or unlike sign according as ^:> + /c — 2 (the 



INTRODUCTION. XIX 

number of changes of sign) is even or odd, or, what amounts 
to the same, accordiyig as 2^ + Jc is even or odd; hence the co- 
factor of Pj, in the new Determinant will be the co-factor of pj, 
in the old with like or unlike sign according as ^9 + ^ is even or 
odd. But^^^is in the first column and first row of the new 
Determinant, hence, by the foregoing, its co-factor is got by 
destroying its column and row, i.e., by destroying the pth col- 
umn and Jcth row of the old Determinant ; hence the co-factor of 
Pj, in the old Determinant is the Determinant left on destroying 
the roiu and column of j)^, but taken -f or — according as p + k 
is even or odd. 

The co-factor of any element may be denoted by the same 
symbol written large ; thus, the co-factor of ^^ is Pj,. 

8. B}' definition, all the terms containing any element are got 
by multiplying that element b}' its co-factor ; if, then, we mul- 
tiply each element of a row (or column) by its co-factor and 
form the sum, we shall get cdl the terms of the Determinant that 
contain any element of that row (or column) ; but evei'i/ term 
contains '0716 element of that row (or column) ; hence we get 
cdl terms of the Determinant; and since no term contains tico 
elements of that row (or column) , we get each term but once. 
Hence, the sum of products of each element of a row (or column) 
hy its oicn co-factor is the Determinant itself ; 

I «i&2C3"- ''^n I = «i^i + ^i-5i + Ci Ci + ••• + '^i^^i, = etc. 

It is to note that the co-factors subscribed 1 contain every 
other subscript in their values hut 1, and every other letter hut 
their own. Xow change the subscript 1 to 2 on both sides of the 
equation ; there results 

I ^'2 ^2^3 • • • ''^n I = «2^1 + &2-Si + C2 Ci H + ^2 JSf-i_. 

The subscripts of the co-factors are not changed, because the 
subscript ^ does not appear in their values. 

iSTow the left side of this equation is a Determinant with two 
rows congruent, namely, the 1st and 2d, since the subscripts of 



XX CO-ORDINATE GEOMETRY. 

the 1st, which were all i were changed to 2, the subscripts of the 
2d ; hence its value is by Art. 6 ; i.e., 

The small letters are the elements of the 2d row, the large 
letters are the co-factors of the corresponding elements of the 
1st row ; plainly the reasoning about columns or about any 
other pair of subscripts would be the same ; hence the sum cf 
products of each element of a row (or column) by the co-factor 
of the corresponding element of any other row (or column) is 0. 

9. If the elements of any row (or column), as the 1st, be 
regarded each as the sutyi of two jpar^-elements, so that 

then we shall have 

The first bracket is clearly the Determinant | aih^c^-'-n^ |, the 
second is | ai"b2Cg •••'^^„ | ; hence, it is plain that 

I (%' + %") ^2^3 •• • ^H I = I Ctl' ^2^3 • • • "^Ki I + I Cty"b2C^ ••'''^n \' 

In this way one Determinant may always be expressed as the 
sujii of two j9a?-^-Determinants which have all their rows (or 
columns) the same as in the whole Determinant, but one pair 
of corresponding ones, while the sum of any two corresponding 
elements in this pair equals the corresponding element in the 
corresponding row (or column) in the whole Determinant. 

It is now clear that any Determinant may be broken up into 
3, or, indeed, into any number of part-Determinants, each ele- 
ment of a row being supposed made up of 3, or any number of 
parts, thus : 

ai = ai'+ai"+ai'", &i'+V+^i"', etc. 

If every row (or column) be broken up into parts this wa}'', 
a part-Determinant ma}' be formed by taking for a 1st column 
any part-column of the 1st column, for a 2d column any part- 



INTRODUCTION. Xxi 

column of the 2d column, and so on throughout. Hence the 
total number of part-Determinants will be the product of the 
numbers of part-columns for all the columns. 



Evaluation of Determinants. 

10. In the Determinant 1 tti^a ^3 ••• ^'nl ^^^ the 2d column to 
the 1st, each element to its correspondent ; we get 

The 2d Determinant on the right has two identical columns 
of 6's, hence its value is ; the 1st on the right is the original 
one ; hence the new Determinant on the left equals the old one. 
If instead of adding the 2d column we had added its ?7i-fold, we 
should have got m times the 2d Determinant on the right, which 
would still be ; plainl}', too, the reasoning about any other 
pair of columns or about rows would be the same. Hence, 
the value of a Determinant is not cliaiiged by adding to each of 
its elements in one row {or column) any fixed multiple of the cor- 
respondent elements in any other roiv (or column). 

This theorem furnishes a readj" method of reducing the degree 
of a Determinant. For by proper additions all the elements of 
a row (or column), say the 1st, may be made but one ; then 
the whole Determinant will be equal to this one multiplied by 
its co-factor, for we shall have 

The degree of this co-factor is clearly one less than the degree 
of the original Determinant ; by repeating this process the 
degree of the Determinant may be brought down to 2 or even 
to 1. 

Thus far the reasoning has been so closely connected and 
withal so simple that it has been deemed best not to interrupt 
it in any way. The following examples will amply illustrate all 
the foregoing. 



xxu 



CO-OEDINATE GEOMETRY. 



1. 4631527. In this permutation the inversions are : 43, 
41, 42, 63, 61, 65, 62, 31, 32, 52, ten in number, the permu- 
tation is even, h ca efg d. Here the inversions are : 5a, ca^ 
ed, fd^ gd., five in number, the permutation is odd. On exchang- 
ing 6 and 2, the number of inversions falls to five, the permu- 
tation becomes odd ; on exchanging a and /, the number of 
inversions rises to eight, the permutation becomes even. 

2. The permutations of 1 2 3 are, in pairs: 123, 132; 213, 
312 ; 231, 321 ; one of each pair is even, one odd. Which? 



3. The Determinant of 2d degree 



tts ho 



4. 



«! 


&1 


Ci 


^2 


h. 


Cs 


% 


h 


Cs 



= (^ih^C'i — «1&3C2 + a2^3Cl 

-ag^sCi. 



-- 0-162 — Of's^i. 
«2^lC3 + %&lC2 



The number of terms is 3 ! = 6, so we write off the combina- 
tion ah c six times and suffix the subscripts permuted. The 
numbers of inversions in the permutations are resp. : 0, 1, 2, 
1, 2, 3, and the signs are prefixed accordingly. A simple 
mechanical rale for calculating the Determinant of 3d degree as 
shown in this diagram : 




The arrows turning up are drawn through the + combina- 
tions ; those turning down, through the — ones. But this does 
not hold for higher degrees. 



«! hy Ci di 



ao &2 Co d-2 



cio bo Co cip 



a^ 64 



di 



= (ly 


ho Co d2 
?>3 Cg dj 

64 C4 c/4 


-h, 
-d. 


Cti) Co Clo 

«3 Cn dn 
(X4 C4 U4 

ag ho c.2 
% h^ C3 

0^4 &4 C4 


+ Ci 



0,2 O2 0/2 

ag O3 dg 
ct^ O4 ct^ 



INTRODUCTION. 



XXlll 



6. 



(h + 0^1 ^1 ^1 


= 


tti bi Ci 


+ 


ai 6i q 


0^2 + 02 62 ^2 




0/2 O2 C2 




ttg &2 C2 


<^3 + «3 ^3 C3 




Clg O3 C3 




0-3 h <^3 



7. 



8. 



9. 



«i 4- ai &i + ^1 Ci 4- yi 

«2 + "2 ^2 + /?2 ^2 + 72 
«3 + Ct3 &3 + ft C3 + 73 

= 1 <^i&2<^3 1 + 1 (hhys 1 4- I ttiftcg I + I aift73 1 + 1 ttidgCs 

H- I ai&273| + I ciift C3 I + I 01^8273 I • 



as + '??^&2 ^2 ^2 
tts + m^g 63 C4 



«1 &1 Ci + 

a-2 ^2 C2 
03 03 C3 

= 1 aj boC^l + m I &162C3I = I tti^sCsl. 



m6i bi Ci 

97162 ^2 ^2 
77263 63 C3 



&1 


h 


Ci 


6, 


62 


C2 


h 


2>3 


C3 



= 6162C3 — 6163C2 4- 62^3 Ci— 6261 C3+ 6361C2— 6362C1 
= 0. 



10. 



«! Ci 61 


= ttiC 


tt2 C2 O2 




«3 C3 63 









= aiC2 63 — C(iC^b2 + ^2^3 61 — tts^i^s + «3Ci62 
— «3C2 6i 



«! 


&1 


Cl 


0.2 


6. 


C2 


«3 


&3 


^3 



11. 



12. 



Also 



4 5 

6 8 



4 


5 


2 


6 


8 


10 


2 


1 


3 



= 32-30 = 2 2 5i=2(16-15)=2 4 5 

3 8 I 3 4 

= 2(16 -15) = 2. 

= 96 + 12 + 100 - 32 - 40 - 90 = 46. 



4 5 2 


= 2 


6 8 10 




2 13 





2 5 2 


= 2 


3 8 10 




1 1 3 





3 


-4 


= 2 


3 


-4 


5 


1 




5 


1 


1 1 


3 









= 2(3 + 20) =46. 



XXIV 



CO-ORDINATE GEOMETRY, 



13. 



2 


3 


5 


7 


= 


-1 


5 


4 


2 


6 




5 


7 


2 


3 


8 




7 


3 


4 


2 


5 




3 



1 


3 


2 


= 


4 


2 


6 




2 


3 


8 




4 


2 


5 





-1 


3 2 


-1 


17 16 


-5 


24 22 


1 


11 11 



14. Reckon 



15. Reckon 



k 


h g 


h 


Jf 


9 


fc 



= -23. 



, and find the co-factors K, H, G, F, (7, J. 



1 


17 


16 


= 


5 


24 


22 




1 


11 


11 





1 


17 


16 


=:z 


2 


77 





79 


77 




1 


27 





28 


27 









5 3 2 


1 


3 


-2 


5 


1 


8 


3 


-2 


1 


a b c 


4 7 8 




6 


4 


-3 




5 - 


-7 


6 




b c a 


9 2 6 




2 


-7 


6 




4 


9 


-5 




cab 



16* Bring the equation 





iC 


^'l^2 


?'2 ^1 


n 


-^2 


r^x^ 


— r3£i'2 


^2 


-'3 



y 

7'l — 9^2 

^2 2/3-^3^/2 
?*2 — ^3 



into the foraa 



?*i rs ^3 


X — 


«/l 2/2 2/3 




1 1 1 





i\ i\ r^ 


y+ 


Xi X2 Xg 




111 





n 


^2 


n 


2/1 


2/2 2/3 


Xj_ 


tVo 


Xq I 



= 



0. 



Multiplication of Determinants. 

11. An interesting case of Art. 9, as illustrated in Example 
7, is this : 



«2«l+^2^1+C2 7lH f-^J'u «2«2+^2^2+- + ^2''2' 






•, anan-\-KPn-\ h^w'-'n 



We notice here that the Greek and the Roman letters enter 
this Determinant in the same way : the Greek appear in the 
columns as the Roman do in the rows, and vice versa. Again, 
this whole Determinant, which we may write /\GB, clearly 
breaks up into the sum of n"" part-Determinants ; for any of 



INTRODUCTION. XXV 

the 1st n part-columns may be combined with any one of the 2d 
n part-cohimns, and so on. But all but n ! of these part-Determi- 
nants vanish; for, if the Eoman letter of one part-column be 
the same as the Roman letter of any other, then on setting out 
the Greek factor the part-determinant will have two columns 
identical, and vanishes by Art. 6. Accordingly, to get part- 
Determinants not = 0, we must pick out each time all the Roman 
letters for our part-Determinant, one for each part-column ; 
hence, on setting out the Greek factors, the part-columns 
become the columns of the Determinant of Roman letters, 
\ CLib^G^ " ' n^\-, which we may write Ai? ; hence, our part-Determi- 
nant contains as a factor a determinant that can differ from 
Ai? only in the order of its columns, i.e., only in sign ; hence, 
being a factor of each part-Determinant, A^ is a factor of the 
whole Determinant. Hence, since the Greek and the Roman 
letters enter ^GR alike, A(t is also a factor of AGB. Now 
the terms of the product A(t' Ai? are of 2 nth. degree, and so are 
the terms of AGR ; also the number of terms both in AGM and 
in A(x ' AM is the same, n ! n ! Hence, AGR can differ from 
AG • AR, if at all, only in sign ; i.e., AGR= ± AG • AR. 
To decide as to the sign, consider the product of the diagonal 
terms ; it is -f in AG ' AR ; the like term is also -f in AGR. 
For it is got b}^ taking the 1st part-column of the 1st column, 
the 2d of the 2d, etc. ; the factors set out are aj^gys ••• ^Vn ^^^^ 
the order of the columns of Roman letters is natural, as in AR ; 
and in AR the diagonal ai^s^s ... n„ is -f- ? ^s is the diagonal term 
in every Determinant. Now one pair of corresponding terms 
being like-signed in AG • AR and AGR^ all are like-signed ; 
i.e., AG'AR = AGR. 
Illustration : 

«iai+&i/?i + Ci7i, ttiag + ^ift-f Ciys, ^icts + ^1^3 + ^lyg 
agai -1-62/^1 + ^271? agag-f 62/?2 + C2y2, Of2a3 + ^2^3 + ^273 
«3 ai + hPi + C3 71, as as 4- h/^2 + C3725 <^3 «3 + hl^3 + ^s 73 

= CLi(32y3 1 ai^sCg 1 +ai72/53 1 CiiC.2bs I + /3i72a3 I biCotts I 

+ ft 0^2 73 I ^i^aCg I -f-7ia2/?3 I Ci Of2&3 I + 7ift as | ^162^^3 | • 



XXVI CO-OEDINATE GEOMETRY. 

Here the inversions in the order of the columns of Eoman letters 
are those got by permuting those letters ; if we restore them to 
natural order, the indices, which are now in natural order, will 
be permuted as were the letters ; also, restoring them will 
change or not change the sign of the term according as the 
number of inversions is even or odd ; but the inversions in 
the order of Roman letters are the same as in the order of the 
Greek ; hence, the Greek products are + or — according as the 
permutations of the letters are even or odd ; hence, the sum 
of the Greek products is the Determinant of the Greek letters. 
As no reference has been made to the number of letters, 3, this 
proof is quite as general as the one already given. Actually 
bringing the Roman letters into natural order, we get 

+ asftyi Ai? - agftyi Ai? = AG^ • Ai?. 

12. If ai= Ai, a2 = A2, ..., l3i = Bi, ..., the Greek letters 
being the co-factors of the corresponding Roman letters, then, 
in ^GB all the elements vanish but the diagonal ones, by 
Art. 8, and these are each AB ; hence AGE has but one term, 
the diagonal term, and that is Ai^**. Writing A for AB and A 
for AG, and remembering AGB — AG > AB, we have 



A • A = A^ or A = A*'-^ ; 

the Determinant of the co-factors of the elements of a Determi- 
nant of nth degree is the (?i — l)th power of this Determinant. 

In particular, if ?i = 3, A = A^. 
If the co-factor of Ai be a^^ and so on, and if the Determinant 
of the co-factors a^', etc., of the co-factors A^^ etc., be written 
A', then 

Now if we multiply ai, 6i, ..., Gg, ..., each element of A by 
^n-2 or A : A, it will be the same as to multiply A by A^"--^", 
since it multiplies each row (or column) by A"~^ ; the resulting 
Determinant is then a*'^"^""*'^ or is A', ^ence, we may con- 



li^TRODUCTION. XXvii 

elude that a' = a-A''"^, a proposition we have not space to prove 
more rigor qu sly. 

In particular, if ?i = 3, a' = a • A. 

Applications. 

13. If there be given a sj^stem of n Equations containiug 
n Unknowns in fii'st degree onh^, it is possible hy successive 
elimination to get one Equation with one Unknown, whence this 
Unknown may be found ; then, by substitution, all the others 
may be found, one by one. But this process is very tedious. 
The theorems of Art. 8 furnish a direct solution of the problem. 

Call the Unknowns u^^ u^^ •••, ii„ ; then 

ci\U^-{-hiU2-\- -\-niitn = 'ki 



If the Equations be multiplied by Ai^ Ao, ••• A^^ in turn and 
summed, where Ai, Ao, •••, A„ are the co-factors of a^, ag, ••• <'^„, 
in the determinant of the coefficients | ai^g^s ••• n^\, the coeffi- 
cient of Ui in the sum will be 

ciiAi + a2^2 H + cin^n-) = 1 <^i ^2 ^3 ' ' ' ''^h |^ ^y Art. 8, 

while the coefficient of any other w, as U2-, will be 

b,A,-^h,A.2-\- ••'-{- KA,, =0, by Art. 8, 

and the absolute term in the sum will be 

7ijAj_ + A-2 A H h K^n = I h hC3 "''i^n\\ 

or, I ai 69 Cs • • • nn \ui = \ h h^Q'" n,, |, 

whence Ui = ' "^ ^ . 

Note that the denominator is the Determinant of the coeffi- 
cients in order ; the .numerator is got from the denominator by 



XX Vm CO-OEDINATE GEOMETRY. 

replacing the column of coefficients of the Unkuown in question 
by the column of absolutes. 

Plainly the reasoning holds alike for all the Unknowns. 

14. If the absolutes, the Zc's, be all 0, then the numerator of 
the value of each Unknown, of each u, has a column of O's, 
hence itself is ; hence the value of each u must be unless 
the denominator also be ; when the Zc's are 0, the Equations are 
Jiomogeneo'us in the it's ; lience 

The condition that n homogeneous equations of 1st degree in n 
Unknowns may consist^ is that the Detei'minant of the coefficients 
of the Unknowns vanish. 

Where the Equations are homogeneous in the u's^ each may 
be divided by one of the w's, say u^ ; the quotients of the u's 
being considered as new Unknowns, neiv u's, there are n Equa- 
tions and only (n — 1) Unknowns, while the coefficients of ?/„ 
are absolutes ; the condition of consistence of the Equations is 
unchanged ; hence 

The condition that n Equations among (n — 1) UnJcnowns in 
1st degree may consist, is that the Determinant of the coefficients 
and the absolutes vanish. 

15. Often it is required to eliminate one Unknown between 
two Equations of higher degree in that Unknown ; or, what is 
the same, to find what relation must hold among the coefficients 
in the two Equations, if the two are to consist, i.e., hold for the 
same values of the Unknown. An example will make this clear. 

Given ax^ -{- bx^ + ex -\- d — and ex^ -\-fx + ^ = ; 
find the condition that these Equations consist, i.e., that the 
roots of the 2d be also roots of the 1st. 

When the first Equation is satisfied, so is this : 

ax^ -{- by? -\- CO? -\- dx = \ 
when the second is, so are these : 

es? -\-fx- -{-gx = and ex"^ +/^ + g^ = ; 



INTEODUCTIOIT. 



XXIX 



hence, when both are satisfied, so are these five: 
O'X^ -\- ax^ -^bccr-\- ex -\-d = 0, 
a-x^-{- hx^ + cx^ -[-dx -f- = 0, 
O-x'^+O-x' + ex- +fx -j-g = 0, 
O-x^^e.'X^ +/x^ -{-gx +0 = 0, 
e-x'^-^fx^ -j-gx^ -^0-x-{-0 = 0. 

Here are 5 Equations containing 4 Unknowns : x'^, x^, a^, x ; 
by Art. 14 they consist when and only when 

a b c d =0. 

a b c d 

e f g 

e f g 

e f g 

Clearly this method is always applicable. Be one Equation 
of ?ith degree, the other of (n + d)th degree ; by multiplying the 
1st by X d times successively we raise it to the (n + <i)th degree, 
and get in all d-\-2 Equations and n-j- d Unknowns ; then by 
multiplying each of the two Equations of {n-\-d)th degree by 
X we get two more Equations and one more Unknown, the next 
higher power of x, x^^'^^^ ; by w — 1 such successive multiplica- 
tions we get in all 2n -{-d Equations and 2n-\-d — l Unknowns ; 
then Art. 14 is to be applied. 

Under the hands of British and Continental masters the 
Theory of Determinants has been of late years built up to colos- 
sal size and applied to almost every branch of mathematics ; in 
fact, it has become well-nigh indispensable to higher research. 
An excellent English work is Muir's Theory of Determinants. 



EXERCISES. 

1. Solve the systems of equations : 

Sx + 4:y — 5z=1, 2x — Sij — 4:z = 9, 4:X—6y + z = 8; 
x — y-{- 22 + 5y = 10, 2x+ '^y — z -\- v = l, 
4?/-3z + 80 — 2u=5, 32-2?/-5a:+ 7y = 3. 



XXX CO-OEDINATE GEOMETRY. 

2. Do these systems of equations consist 1 

6x—Sy = 7, 8x-\-d7j = 9, 3x — 2j/ = — 4; 

2x -{■ 3i/ — 4:z= 5, L>x — 2y -{- 3z=7, x -\- 5y — 6z= — 

4:x + Sy-Bz = 2. 



CO-OEDINATE GEOMETEY. 



-oo^&ioo- 



Part I. The Plane. 



CHAPTER I. 

INTRODUCTORY: FIRST NOTIONS. 

Function and Argument. 

1. In a table of logarithms are found two series of corre- 
sponding values : one of natural numbers and one of logarithms. 
Given any number, we may find from such table a corresponding 
logarithm ; given any logarithm, we may find tJie corresponding 
number. Like may be said of a table of natural sines : given 
any number (expressed commonly in degrees) , we can find the 
corresponding sine; given any sine, we can find a correspond- 
ing number. Such tables are calculated to a greater or less 
degree of exactness by rules or formulae ; other like tables are 
found in works on Physics, calculated, however, not by rule, 
but by experiment. From such a table we may find (within 
certain limits) e.g. the tension of saturated vapor of water for 
every degree of temperature, and conversely ; but we know no 
rule to calculate one from the other. 

Two magnitudes^ such that to any value of one corresponds a 
value of the other., are ccdled functions of each other. 

Such are a number and its logarithm ; a number and its sine ; 
the surface or volume of a sphere and its radius ; the velocity 
of a wave-motion, as of sound, and the elasticity of the medium ; 
the density of pure water and its temperature ; etc. 



2 CO-OBDINATE GEOMETRY. 

2. As is seen, in Physics the interdependent magnitudes 
called functions in general define physical states, and the nature 
of their interdependence cannot generall}- be stated in a formula 
or rule ; in Mathematics the interdependents are empty forms, 
symbols : x, y, z, a^ b, c, and the nature of their interdepend- 
ence is expressed by Si formula or equation. 

Take, for example, the Eq.,* 2x-\-^y = 12. 

Assigning arbitrary values to x respecti'vely y^ we reckon the 
corresponding values of y resp. x by the fo7'mulce (rules) : 



y=12 — 2x:3 resp. x=12 — Sy:2. 

A series of pairs of corresponding values is 
{x,y)=\{-3,6); (-2,-1/); (-1,^^) ; (0,4) ; (1,J/) ; 
(2,1); (3,2); ... |. 

In the unsolved Eq., 2x-^3y=12, x and y stand on 
precisely like footing : each is an implicit function of the other, 
but in the Eq, solved as to one of them, say y, thus 



y=12 — 2x: 3, 

they no longer stand on like footing ; contrariwise, this Eq. 
gives a ride for reckoning the value of y for any assigned value 
of a;, but 7iot conversely. That symbol to luhich ice assign arbi- 
trary values is called the Argument ; the symbol whose values 
are reckoned is called specifically the function; if the Eq. be 
solved as to any symbol, that sj^mbol is called an explicit 
function. 

Illustration, x^ -\- y^ = 25 : here x and y are each an implicit 
function of the other ; solved. 



y = ± V25 — x^ : here y is an explicit function of the argument x. 



x= ± V25 — ?/- : here x is an exjMcit function of the argument y. 



* Short for Equation. 



KINDS OF rUKCTIONS. 3 

N. B. Though the relation between two symbols be accurately 
expressed b}' an Eq., yet it is not in general possible to state a 
rule for reckoning one through the other, since the general Eq. 
of degree higher than the fifth has not yet been solved. None 
the less, the symbol to which we suppose arbitrary values 
assigned is still called the argument ; that, as to which we su[)- 
pose the Eq. solved, the function. 

3. If in the Eq. connecting two S3'mbols each be operated 
upon by 2i finite number of algebraic operations, additions, mul- 
tiplications, involutions, and their inverses, then is each called 
an algebraic function of the other ; but if the number of such 
operations upon either be infinite^ then are the}' called transcen- 
dental functions of each other ; as 



/%-»0 rpO /y%i 

.V=sinx = ct'-|--h^-^4-- =t {-ly 



„2n + l 



3 ! 5 ! 7 ! n=o 2?i+l ! 

/yi2 r,^ n-A n= oo ^n 

^ 2 ! 3 ! 4 ! n= n ! 

More important for us is this distinction : when to one value 
of the argument correspond one, two, three, resp. many values 
of the function, the latter is called a one-, tico-, three-, resp. 
many-yalned function of the argument ; as 

2x -[-Sy = 12 : x and y each o?2e-valued functions of 

the other ; 
x^ -{- y'^ = 1-^ '. X and y each ^tco- valued functions of 

the other ; 
y- = ^px : £c a 07ie-valued function of y, y a, two- 

valued function of x ; 

y = COS x — V ( — 1 ) " - — 1 y a one-xalued function of x, x an in- 
''^^^- finitely many -\[dued function of y 
(since, if x= r be an}' root of the Eq. for any assigned value 
of y, then is also x = ±2mT±r a root, n being an}' natural 
number) . 



CO-ORDIISrATE GEOMETRY. 



EXERCISES. 



What functions of each other are x and y in 

2 2 

— |- "/- = 1 ; y^ — S axy + x^ = ; [x"^ + y'^Y — 4a'^x^j/^ ; 
d^ IP- 



From the last Eq. express x as an explicit function of y, 

4. To different values of the argument correspond in general 
different values of the function ; if the difference between two 
argument-values be very small, in general the difference be- 
tween the corresponding function- values will be very small : as 
the argument changes gradually, so does the function. Be 
now ^2 — a?! ^ Aaj the difference between two argument- values, 
2/2 — 2/i = ^y the difference between the corresponding function- 
values ; if ;then., by taking Acc<o-, we can make A2/<cr' for all 
valuesQf Aa?< a-, where the o-'s mean magnitudes small at will, 
then is y called a continuous function of x. 

Illustrations. ^Ix -\-Zy =^V1. If {x^^y-^^ (^2^2/2) he two 
pairs of corresponding values, 

then 2 iCi + 3 y^ =12, and 2 cCs + 3 ?/2 = 12 ; 

whence ccg — ^1 = — f ( ^2 — Vi) 5 or A^/ = — -| Ao?. 

The sign — shows that, as either x or y increases, the other 
decreases ; but we have here to deal only with the absolute (or 
signless) values of the differences Ax and A?/, and since their 
ratio 2 : 3 is finite, clearh' we can make and keep either small 
at will by making and keeping the other small at will. This is 
so for every finite value of x or y ; hence, each is a continuous 
function of the other for — oo<cc<oo, — oo<?/<cc. 

Like may be shown of the functions y = sin x, y = cos x^ but 

not of their quotient 

, sin X 
y = tan x = . 

cos a? 



KINDS OF FUKCTIONS. 



For 07 = [-7i, 7i very small, sin x = —l nearly, cos x 

is + and = nearly ; hence, y = tan x is very great and neg- 
ative. As li increases toward -, since increases toward — 
(i.e., nears from the — side), cos a? increases toward -[-1, 
tanoj increases toward — gradually ; as li nears tt, x nears -, 
sin a; nears -|-1, coscc nears +0, tan cc becomes -f- and very 
great, but changes throughout gradually with x. For cc = - — cr, 

(T small at will and -f^ tan x is very great and + ; for 
£C= - + 0-, coscc changes from the -f to the — side of 0, since 

changes not, tancc changes from being very great and + to 

being very great and — . Hence, for — -<x<-, tancc is 

a continuous function of x ; but for o^ = - , tan a; is a discon- 

2 

tinuous function of xi if cci < - , a?2 > - 5 we can ?ioi mal^e 
2/2 — 2/i = tan ct'a — tan .Tj = A?/ small at will by making x^—x-^ 
= Aa? small at will ; as x passes through the A^alue a?= - , tancc 

springs from -\-oo to — 00 . The value cc = - is called a ^om^ 



of discontinuity for the function 2/ = tan x. Since tt is the period 
of the tangent, i.e., tan 07 = 

also a point of discontinuity. 



of the tangent, i.e., tana7= tan(aj ± tt), x=z(^±2n -\-1)- is 



EXERCISE. 

Show that 7/ = c . is discontinuous for x =: a. 

J_ 

gX - « J- 1 

Hint. As x nears a, increasing, y nears — c; as a: nears a, decreasing 
y nears -j-c; as a; passes through a, increasing resp. decreasing, y springs 
from — c to +c resp. from 4-c to — c. 



CO-OEDINATE GEOMETRYo 



Geometric Representation of Magnitudes. 

5. Any measurable magnitude may be represented by a 
number, called its metric number: the ratio of the magnitude 
to an assumed unit-magnitude ; a number may be represented, 
or pictured, by some definite part of a line, some arbitrarj' tract 
(i.e., definite part of a right line) being taken to picture 1. 
Thus, if AB picture 1, BD will picture 2, BC will picture V2, 




h" 



C" c c 



AS, 

BCD will picture ir. Instead of the tract whose metric number 
is a may be said briefly the tract a. On the ray OD lay off 
from any number of tracts picturing as many (positive) 
numbers. To picture the sum of any two numbers, a and 6, 
lay off" a tract s equal to the sum of the tracts a and 5. To 
picture the difference of two numbers, a and 5, lay off a tract d 
such that the sum of the tracts h and d shall equal the tract a. 
To do this, lay off from the end^ of tract a, toward 0, i.e., 
counter to the direction a was laid off in, a tract 6 to C; then is 
OC the sought tract d. Three cases may arise : 

(1) ay-b^ then the point C fails between and A ; 

(2) a = 6', then the point C" falls on 0, the difference d is ; 

(3) a<Z>", then the point C" falls beyond leftward; but 
then the difference a — b" or d is negative ; hence negative 
numbers are pictured b}' tracts laid off counter to the direction 
in which are laid off tracts picturing positive numbers. But in 
this way was laid off a positive tract, to subtract it ; hence, to 



PICTURES OF NUMBERS. 7 

add a negative tract, subtract an equal positive one ; hence, too, 
to subtract a negative tract, add an equal positive one. 

N.B. To add to a tract, or to subtract from it, we lay off 
from its end: forwards, from it, to add a positive, or subtract 
a negative, tract ; baciiwards on it, to subtract a positive, or add 
a negative, tract. Like reasoning and conclusions hold for 
angles and arcs. 

6. Assuming, then, any RL. (short for right line), we may 
picture all real numbers b}' tracts laid off on it rightward and 
leftward from any assumed point 0. As all such tracts have 



a A 

the same beginning, 0, each is full}' defined by its end. Ac- 
cordmgl}', not only the tract OA by its length and direction, but 
its end, the point ^, by its position as to 0, pictures the num- 
ber a ; so, every point on the RL. OD pictures some real 
number, the RL. itself pictures the whole of real numbers. 

If we take two RLs., as OX, OY^ intersecting under any ^ w, 
each will picture by its 
points the whole of real 
numbers, the section 
naturally taken to pict- 
ure zero in each picture. 
Any point P in the plane 
of the RLs. will pticture 
then not simply one 
number but a pair of 
numbers, as (a, b) ; for 
its distances from the zero-point 0, measured along these RLs., 
are a and b. 

N.B. The choice of RLs. and of positive and negative direc- 
tions is arbitrary ; it is common to treat rightward and upward 
as -{- , leftward and downward as — . 




CO-ORDINATE GEOMETRY. 




EXERCISES. 

In the figure, the corner-points picture the pairs of numbers bracketed 
by them. 

1. Find the points that picture these pairs : (1,3), (— 3, — 1), (— 2, 1), 

(4,-5), (0,1), (0,6), (5,0), 
(6,0), (0,0). 
C-2,2) / 

2. "Where are all points 

picturing pairs whose first 
term is ; i.e., of the form 

3. Of the form (a, 0) ? 

4. Where are all pict- 
ures of pairs whose terms 
are equal ; i.e., of the form 

5. Of the form {a, — a) "? 

6. Where are all pictures of pairs whose first terms are all alike, 
second terms unlike ? 

7. Whose second terms are all alike, the first unlike 1 

7. Be now f{x^ y) = ^ (read : the /-function of x and y 
equals 0) any Eq. determining x and y as functions of each 
other, and suppose the functions continuous and one-valued. 
Be (Xi, 2/i), (^21 2/2)5 • • • (^nt Vn) pairs of corresponding values ; 
i.e., be/(iri,2/i) = 0, /{x^^y-i) = 0, . ../{x^.y,,) = 0. Each pair is 
pictured by a point in the plane of OX and OY. Suppose the a;'s 
subscribed in the order of size, thus : cci< iC2 < ^3 < • • ' < ^w ^J 
taking consecutive values of x very close to each other, we make 
the consecutive values of y very close to each other ; this series 
of pairs of values of x and y will then be pictured b}- a series of 
points consecutively very close to each other. Now, it is true, 
however close together we may heap these points, we can never 
make out of them a line. But if the function be continuous, by 
taking x^_^i — cc^^ = o- we can make 2/&+1 — Vk = ""'5 ^^^^l for every 
value of £c between x,,^i and Xf, the corresponding value of y will 
differ from y^ by < o-', positively or negativelj^ ; i.e., all points 




DETEEMINATION OF POSITION. 9 

picturing pairs of values for x between Xj,j^^ and Xj^ will lie in the 
double parallelogram whose sides are o- and 2cr'. Hence all 
points picturing pairs of values that satisfy the Eq. /(a;, y) = 
lie in a series of con- 
tiguous double parallelo- 
grams, whose sides are 
small at will ; so, too, 
are the diagonals of the 
single parallelograms. 

The train of diao-onals 
(that one of each couple 
being taken that joins 
two picturing points) will 
form a polygon (open or closed) whose vertices are picturing 
points, and by taking the values of cc, and therefore of ?/, ever 
closer and closer together, we make this polygon near as its limit 
a definite curve, of which each diagonal is a chord. Every pair 
of values satisfying the Eq. /(a?, ?/) = is pictured by a point 
on this curve ; and conversely, everj' point of this curve pictures 
a pair of values satisfying the Eq. /(x, y) = 0. Hence, 

A geometric pictiire of the Eq. fix., y) =0 is a plane curve. 
We say indifferently the Eq. /(o.^, ?/) = and the curve 
/(^,2/)=0. 

N.B. The curve breaks up in case : of a man^'-valued function, 
into many branches ; of a discontinuous function, into distinct 
parts. But the reasoning needs but slight change. If some 
other than a RL. be used to picture a series of numbers, like 
reasonino- and conclusions hold. 



'» 



Determination of Position on a Surface. 

8. We say of a surface : it is doubly extended, or has two 
dimensions, meaning that two independent measurements are 
necessary and sufficient to fix any point on the surface. Thus 
we know any place on the earth's suiface, knowing its latitude 
and longitude. In this familiar example we suppose the sur- 



10 



CO-OEDINATE GEOMETRY. 



face covered with a double system of lines : half-meridians and 
perpendicular parallel small circles. Through an}' point of the 
surface passes one and only one half-meridian, one and only 
one parallel ; hence, knowing an}' point, we know its meridian 
and parallel. Also, any half-meridian cuts any parallel in oue 
and only one point ; hence, knowing any meridian and any par- 
allel, we know their junction-point. The parallels are named 
from their angular distance from the mid-parallel (equator) ; 
the half -meridians from their angular distance from an assumed 
fixed half-meridian. 

Likewise we may think a plane covered with a double system 
of (say parallel right) lines, making any angle w with each 
other. Thi'ough any point in the plane passes one and only 

one line of each system ; 
hence, knowing any point, 
we know what pair of lines 
meet in it. Conversely, any 
pair of lines meet in one and 
only one point of the plane ; 
hence, knowing any pair of 
lines, we know their junc- 
tion-point. We name and 
know each line of a system 
by its distance from an 
assumed fixed Ihie of the 
system (measured on any line of the other system). If this 
measurement be rightward or up, the metric number of the 
distance is marked + ; if leftward or down, — . The 
assumed fixed RLs., as OX, OF, are called Co-ordinate Axes, 
or axes of X and Y, or X- and F-axes. The angle w, reck- 
oned from the + X- to the + F-axis, is called the co-ordinate 
angle ; for w = 90°, the axes are rectangular ; otherwise, oblique. 
Tlie junction-point, 0, of the axes is called the Origin. 

9. A point is fixed as the junction of a pair of parallels, or 
co-ordinate lines ; conversely, a pair of co-ordinate lines are 




POLAE CO-OHDINATES. 11 

fixed as meeting in a point. The distance from the origin 
at which a co-ordinate line cuts the X- resp. Y-axis is 
called its intercept on that axis. Such intercepts are denoted 
by the symbols x resp. y. If a be the metric number of the 
ic- intercept of any parallel to the y-axis, then is this par- 
allel known completely from the Eq. x = a, which is there- 
fore called its Eq. So y = b is the Eq. of a parallel 
to the X-axis making an intercept b on the I"- axis. The 
junction-point of this pair is known completely from the two 
Eqs. x—a, y = h, which are therefore called the Eqs. of 
the point. The point itself is spoken of as the point (a, b) or 
as P (a, b) ; a is called the abscissa or x of the point, b, its ordi- 
nate or y ; a and 6, its co-ordinates or its x and y. We may 
now convert the proposition of Art. 6, thus : Any point in a 
plane may be represented by a pair of numbers : the rectilinear 
co-ordinates of the point. 

10. We may think the plane covered with some other double 

system of lines : as a s^'stem of rays from the centre of a sys- 
tem of concentric cir- 
cles. Each point is 
fixed as the junction 
of a pair of co-ordi- 
nate lines : ray and 
circle ; each pair of 
such co-ordinate lines 
is fixed as having 
such a junction-point. 
Each ray is known 
and named from its 
angle with some assumed fixed ray, as OD, called base-line or 
polar-axis; angles reckoned clocktvise, as 0', are marked — , 
those reckoned coiinter-docktvise, as 0, are marked -f ; the 
direction of a ray which bounds the ray's angle ^ as OP, is taken 
+ ; the coim/e?*-direction, as OP", — . Each circle is known 
and named from the radius p. The angle ^ of a ray and the 




12 CO-OEDIKATE GEOMETRY. 

radius p of a circle are called tlie polar co-ordinates of tlie 
junction -point of ray and circle ; i.e., of the point {p,0). p = c 
is the Eq. of a circle : it declares that every point of the circle 
is distant c from ; =d is the Eq. of a ray : it declares 
that the radius vector from to any point of the ra}' is sloped 
d to the polar axis. Ray and circle meet in the point whose 
Eqs. are p= c, = d, i.e., in the point (p, 0). 

N.B. To fix all points in the plane by rectilinear co-ordinates, 
X and ?/, it is necessary to let each range in value from — oo to 
-f- 00 ; but it is sufficient to let p range from to -{- oo , from 
to 27r. So confining p and we have but one pair of values 
to fix any one point ; but if we let each range from — oo to H-oc, 
then any point having co-ordinates (p, 0) will also have co- 
ordinates ( — p,0-\-7r), (— p, — 7r + ^)i (p7— 2 7r + ^), and in 
each of these four pairs we may suppose increased or de- 
creased by 2 nrr., n being nnj' natural number. In polar co- 
ordinates with this range, to any pair of values corresponds 
but one point, but to any point correspond four injinities of 
pairs of values. 

11. B}^ reasoning quite like that of Art. 7 it may now be 
shown that the geometric picture of any Eq. between polar 
co-ordinsrtes, as </>(p5 0) =0, is a plane curve. Every 
point of the curve pictures a pair of values of p and ^ satis- 
fying the Eq. 4>{pi ^) = *^ I conversely, every pair of 
values of p and satisfying the Eq. <;^ ( p, ^) = is 
pictured by a point of the curve. We speak indifferently of 
the Eq. ^ (p, (9) = and of the curve ^ (p, 0) = 0. 

12. There are various other kinds of co-ordinates, as bi- 
polar, trilinear, homogeneous, elliptic ; but rectilinear (called 
also Cartesian, from Descartes, the inventor) and polar are 
the most common and important. 

A point that may be anywhere in a plane and a pair of co- 
ordhiates that may have any values may be said to have tico 
degrees of freedom ; a point that may be an3'where on some 



CO-ORDINATES. 13 

curve in a plane and a pair of co-ordinates whose values must 
satisfy some Eq. may be said to have one degree of freedom ; 
a point that must have one of several definite positions and a 
pair of co-ordinates that must have one of several definite sets 
of values may be said to have no degree of freedom. Thus 
it is seen that mobility in the point corresponds to variability 
in the pah' of values. 




It is to be noted that the same pair of values will in general 
be pictured by different points, not only in systems of different 
co-ordinates, as rectilinear and polar, but also in different 
systems of the same co-ordinates. Thus the pair (2, 1) is 
pictured by the point P in the S3'stem OX, Y, but by the point 
P in the system O'X', O'Y'. Conversely, in different co-ordi- 
nate systems the same point will picture different pairs of 
values. See Art. 21. 

EXERCISES. 

Assume a system of rectangular axes, also take tlie + X-axis as a 
polar axis ; then, 

1. Eind the point I p = 2, 6=—\, and show that its rectangular co-ords. 
are(\/3,l). V ^J 

2. Show that the rectang. co-ords. of {p, 6) are x = p cos 6, y — psin 0. 

3. Hence, express p and Q through x and y. 

4. Eind the rectang. and polar Eqs. of the axes and of a circle about 
0, radius 5. 



14 CO-OKDINATE GEOMETRY. 

Summary. 

13. The results reached so far may thus be summed : 

A. A pair of numbers (as corresponding values of argument 

and function) may be pictured geometrically by a point in 
a plane. 

A'. A point in a plane may be pictured algebraically by a pair 
of numbers (the co-ordinates of the point) . 

The point picturing a number-pair and the number-pair pic- 
turing a point will vary with the co-ordinate system chosen. 

B. An equation (or functional relation) between tivo symbols 

(as /(cc, y) =0, (/)(p, ^) , may be pictured geometrically by 
a plane curve. 

B'. A plane curve may be pictured algebraically by an equation 
between two symbols (called current co-ordinates of a 
point of the curve) . 

The curve picturing an equation and the equation picturing 
a curve will vary with the co-ordinate system chosen. 

Strict proof of B' is neither in place nor needed here ; as 
occasion may offer it will be verified as we go on. 

14. The doctrine based on these facts is named Co-ordinate 
(or Analytic, or Algebraic) Geometry of the Plane. 

Its problem is tvfofold : 

I. Given any algebraic form (as /(a?, y) =0), to picture it by 

a geometric form in a plane (a plane curve) and to inter- 
pret its properties as geometric properties (of the curve) . 

II. Given any geometric form in a plane (a plane curve), to 

picture it by an algebraic form (as f{x^ y) = 0) and 
thence deduce its properties cdgebraically . 

N.B. Not to repel by over-subtlety, by a number has thus 
far been meant a (so-called) real number. But, as the student 



SUBJECT DEFINED. 15 

may know, there are equations whose roots (some or all) are 
(so-called) imaginary numbers, i.e., numbers iuTolving the 
sj'mbol i or V — 1 ; as x^-{-y^= — l. This Eq. is satisfied by 
no pair of real values of x and y ; hence it cannot be pictured 
geometrically in the plane of the axes OX, 07", in which, e.g.. 
x--\-y-=l is pictured by a circle about with radius 1. 

since every point of this plane pictures and pictures only a 
pair of real values of x and y. If, then, x^-\-y-= —1 can 
be pictured geometrically at all, it must be by some geometric 
form not in the plane of OX, OY. The whole question of the 
depiction of pairs of imaginar}^ numbers must be reserved. 

A 7'eal number may be defined as one whose second power is 
positive ; an iina.ginary^ as one whose second power is negative ; 
a complex number, as made up of a reed and an imaginary 
part. 



i:j 



CO-OKDINATE GEOMETRY. 



CHAPTER 11. 

THE RIGHT LINE. 

Before attacking the problem proper of Co-ordinate Geome- 
try it may be well to establish certain useful elementary 
relations. 




15. Distance between two points in terms of their rectilinear 
co-ordinates. Let the points Pi, P^ picture the pairs written 
beside them. Then, at once, 



dj^ = X2 — x{ + 112 — y{ — 2 0^2 — ^1 • 2/2 — 2/i • cos (tt — u)) , 



or, d? = X2 — x{ + 2/2 — 2/1 + 2 0^2 — ^1 • 2/2 — 2/1 • cos w. 
N.B. We may as well write x^ — x^ and 2/1 — 2/2- 

Corollary 1. For w = 90°, cos w = ; 



.' . d" — X2 x-^ "r 2/2 2/1 5 I.e., 

The squared distance betioeen two points equals the sum of the 
squared rectangular co-ordinate differences of the points. 

CoR. 2. If one of the points, as P^, be the origin, then 

^'1 = 0, 2/1 = 0; .•.d^ = a.v + 2/2' + 2.T22/2COSC0; i.e., 



INTERSECTION OF CURVES. 17 

The squared distance of a point from the origin equals the sum 
of its squared rectilinear co-ordinate differences plus tivice their 
product by the cosine of the co-ordinate angle. 

N.B. It will be convenient to use certain self -explaining 

symbols and abbreviations : as, jL for perpendicular ; A for 

triangle ; ^ for angle ; L for right angle ; II for parallel ; Eq. 

for equation ; Cd. for co-ordinate ; RL. for right line ; P (x, y) 

or simply P, or simply '(ic, y) for the point whose co-ordinates 

are x and y ', Pi or (cci, y^) for the point whose co-ordinates are 

x^ and ?/i ; and so for other subscripts, the uniform use of 

I 1 

the subscript being to limit a general symbol; also P1P2 or 

1 1 

(a^i, 2/2) (a^2? 2/2) for the tract from Pi (Xi, 2/1) to Pg {x^^ 2/2) • 

EXERCISES. 

1. If {pi,Qi), {p'l^^-i) ^G two points, 5 their distance apart, show that 
S2 = p,2 + p^^ _ 2 pi p^ cos e^-e.,. 

2. The vertices of a A are (2, 4), ( — 2, 7), (—6, — 8) , draw it, and find 
lengths of its sides, for co = 90°, and for w = 60^. 

3. Draw the 4-side (quadrilateral) whose vertices are (7,2), (0,9), 
(—3,-1), ( — 6,4), and find lengths of sides and diagonals, for&; = 90°, 

and on = 45°. 

4. Eind length of tract between (17, 30° 11') and (19, 48° 26'). 

5. Find points on Y-axis distant d from {x^,ij-^), for co = 90°. 

6. Say (by an Eq.) that (z^, y^) is distant 11 from (7, —2) ; co =60°. 

7. Say that (x, ?/) is equidistant from (2, 5) and (—11,1) ; a) = 45°. 

8. Find {x, y) equidistant from (2,-13), (-9,5), (17,23), a; = 90°. 

9. The tract (5, - 3) (22, y) is VSU long ; find y, for co = 90°. 
10. When is (4, 5) equidistant from (—3, 1) and (9, — 2) ? 

16. If the Cds. (x', y') satisfy an Eq. f {x, ?/) = 0, then is 
the point (x', y') on the curve fi (x, y) = ; if the same pair 
satisfy a second Eq. /g (x^ ?/) = 0, then the same point (x', ?/') 
is on a second curve f (x, 2/) = ; conversed, if (x', y') be a 
common, or junction, point of two curves : f (x, y) = 0, 
/2(^5 2/)=0? then the pair (a?', ?/') satisfies both Eqs., i.e., 



18 CO-OKDINATE GEOMETRY. 

fii^'^y') = and /2 (x'^ y') = 0. Hence, to find the Cds. of 
the junction-points of two curves, solve their Eqs. as simultaneous. 
From Algebra we know that the solution of two simultaneous 
Eqs., one of pth and one of 5th degree, involves in general the 
solution of one Eq. of j9f^th degree ; such an Eq. haspg roots ; 
therefore, there will be in general j^^g pairs of values of x and y 
satisfying both Eqs. ; i.e., two curves, one of pth and one of 
gth degree, meet in pq points. Two, three, or many pairs of 
values may be equal, in which case there is a double, triple, or 
multiple common point ; two or 2 n pairs may be imaginar}^, 
where two or 2 n common points are imaginary, not in our 
plane of OX, OY. 

Illustrations. 1. 3x—7y = 55 and 5 a? -f- 2?/ = — 4 meet in 
(2,-7). 

2. x^ -{- (d — y) (o — ^x — y) = and x-{- y=7 meet in (4,3) 
and (-2,9). 

3. 9x^-\-10xy +y^= 273 meets 9x^ - lOxy -\- y^= 33 in 
(1,12), (-1,-12), (4,3), (-4,-3). 

4. y^ = 4:X, x^= Ay ; hence, ^/"^ — 64?/ =0 ; 

or ?/(?/ — 4) (?/- + 4?/ + 16) = ; common are (0,0), (4,4), 
(_2-i2V3, -2 + 12V3), (-2 + 12V3, -2-i2 /3). 

The last two pairs cannot be pictured by points in our plane ; 
in what sense they are section-points of the two curves in our 
plane cannot now be made clear. 

On review let the student construct the above curves. 



17. Cds. of the point that divides a tract P1P2 in a given 
ratio /xi'. fjio. 

If P(x,y) be the point, so that 

P^P:PP, = (.,:ix,, 

then, by similar A, x — Xi: X2 — x =■ jxi'. 1x2', 

/Xj %)Uo "Y" /X9 *^\ 

or, X = 1 ; 



THACT-DIVISIOX. 19 

Ml 2/' ~\~ l^'> Vi 
likewise, y = "^ — Notice the order of the subscripts. 

/^l "T" /^2 

If either term of the ratio^ as /xg, be negative^ then is P'Pa to 
reckon counter to P'Pi ; i.e., P' falls without the tract, next to 
Pg. The division is then called outer. Conversely, if the 
division be outer ^ one term of the ratio, and hence the ratio 
itself, is negative. 

The formulse yield only one pair of values of x and y ; 
hence, onl}^ one point divides a tract in a given ratio. 




PiP and PP2 (or P^P' and P' Po) are called segments of the 
tract PiPo, and P^PiPP, (or P^P'-.P'P,) is called the 
distance-ratio of P (or P') to Pi and P^. This distance-ratio is 
-f- or — according as the division is inner or outer. 

If P be the (inner) mid-point of the tract, then /.ij = /xg, 



.•.x = Xj^-\-X2:2, y = yi + yo:'2 ; i.e., 

^/ie CcZs. of the (inner) mid-point of a tract are the half-sunns of 
the like Cds. of its ends. 

It fxi = — /xg, P' is the outer mid-point, and x and y are 
infinite ; the outer tnid-point of a tract is at 00 * on the RL. the 
tract is part of. 

EXERCISES. 



1. Find the Cds. of the points which divide the tract (7, 11) (3, 5) in 

I 1 

the ratio 2 : 3, and the tract (3, 13)(— 7, — 1) in the ratio 3 : — 4. 

* See note, page 196. 



20 CO-OKDINATE GEOMETRY. 

2. The vertices of a A are {t-^,^Ji), (.r^, i/^), {^3, Us) ', find the points that 
cut its medials in the ratio 2 : 1, reckoning from the vertices. 

Hint. Take care to think as much, and reckon as little, as possible. 
Here, taking any vertex, we find the x of the division-point on the medial 
is Xj + ^2 + X3 : 3 ; .-. the ;(/ is ^i + ^2 + i/s • ^- These expressions, being sym- 
metric, like-formed, as to the subscripts, hold for all the medials; .-. the 
3 points fall together, are one. This point is named mass-centre of the A. 

3. A point P starts from {x-^^,y-^) and moves half-way toward {x.^,y^), 
then turns and moves onQ-tldrd of the Avay toward (^3, 3/3), then one-fourth 
of the way toward (^4, ?/i), and so on, till at last it moves one 7ith of the 
way toward (xn, y-n) ', where does it stop 1 

The final position of P is called mid-centre or mean point of the n points. 

4. The point P starts from P^ and moves over ^ of the way toward 
P^, then over ^'•^ of the way toward P^, then over ^^ of the 

H-1 + l^-i A'l + /"2 + /^3 

way toward P^, and so on ; where does it stop 1 

The final position of P is called centre of proportional distances. 

5. Three vertices of a parallelogram are (.^p ?/i), {^i,y2)> (^35^/3) ; find 
the Cds. of the 4th and of mid-points of the diagonals. 

What do the results mean geometrically 1 



Parallel Projections. 

18. The intercept^ on any RL.^ made by two \\ planes through 
the ends of a tract is named parallel projection of the tract on 
the RL. If the planes be -L to the RL., the projection is 




orthogonQd ; otherwise, oblique. Thus, OR and OD are projec- 
tions of OB : OR^ orthogonal ; OD, oblique. 



PARALLEL PROJECTIONS. 21 

Clearly, projections of the same tract by the same planes on 
11 RLs. are equal. Accordingh^, in comparing the lengths of a 
tract and its projections, we may suppose all the lines of pro- 
jection to pass through one end of the tract. Calling the tract 

^, its projection p, and denoting by dcV the angle from any 
direction d reckoned around to any other direction d\ we have 
by the Law of Sines, 

. p :t = sin tl : sin|9/, whence p = t ~, 



sin pZ 
i.e., tJie projection of a tract equals the product of the tract and the 
quotient of the sine of the slope of the tract to the direction of pro- 
jection by the sine of the slope of the projection to the same 
direction. 

By odds the most important H projection is the orthogoncd. 
The ^ of a tract with a line of orth. proj. is named directioii- 
angle, its cosine is the direction-cosine of the tract. Since the 
^ of a RL. with a plane is the complement of its ^ with a _L to 

the plane, we have p = t-cosp)t; i.e., orth. proj. of a tract 
= product of the tract by its direction-cosi7ie. 

19. If we project on any RL. the sides of any closed poly- 
gon, taken in order, the end of the projection of the last side 
will fall on the beginning of the projection of thej^rs;^; i.e., 

The sum of the projections of the sides of a closed polygon 
is 0. 

Hence, The projection of any side equals the negative sum of 
the projections of the other sides. 

Or, The sum of the projections of a train of tracts between tivo 
points equcds the projection of the one tract between them. 

If the tracts projected and the RLs. they are projected on 
lie all in one plane, we may put projecting RLs. for projecting 
planes. For this, the figure illustrates the above proposi- 
tions. 



22 



CO-OEDINATE GEOMETRY. 



We see A'B' : AB = sin A'BB' : sin AB'B. If AB'B = 90°, 
A'B' == AB . cos BA'B'. A'B' is tlie projection botli of AB and 
of AFEDCB; the projection of BCDEFA is ^'^' or -A!B'. 




20. The tract from the origin to any point is called the 
radius vector of that point. In the light of the above we may 
now define : 

The rectilinear eels, of a point are the projections of its radius 
vector on each of two axes in its plane .^ \\ to the other axis. 

The polar Cds. of a point are its radius vector and the direc- 
tion-angle of its radius vector reckoned from the polar axis. 

The doctrine of projections is of prime importance in mathe- 
matics. It is here used to treat the 



Transformation of Co-ordinates. \ 

21. The Cds. of a point vary with the system of Cds. 
(Art. 12). To express the Cds. of a point, in one S3'stem, 
through the Cds. of the same point, in another s^'stem, is to 
transform the Cds. 

Several cases arise. 

I. To pass from rectilinear to polar (7(^s.,the origin being the 
same for both. From the above definitions, or from the figure, 
we have at once : 

x: p= sin py : sin xy = sin ((d — 0) i sin w, 
2^ : p = sin xp : sin xy=: sin : sin w ; " 



TEANSFORMATION OF CO-OEDINATES. 



23 



sin(aj — ^) sin^ 

sin oj sm CO 




These Eqs. presume that the X-axis is the polar axis ; if the 
X-axis be sloped a to the polar axis, put ^ — a for ^. 

Foraj = 90°, x = pQo^O, p — psiuO. 

II. To x>ass from one rectilinear system to another loith same 
origin. By Art. 19 the proj. of p, on 0X\\ to OF, equals the 
sum of the projs. of x' and ?/' ; 

'y 




, sm x'y , , sm w . ' ^ r • ; . r • / 

..x=^x'' ~-^y ' '-^-^ ', or, X'^mxy =x^^iTix'y-\-y' smyy^ 

sin xy sin xy 

/-> ^—^ /~^ 

and y « sin xy = x' sin xx' -f y' sin xy'. 



24 



CO-OKDINATE GEOMETIIY. 



This last Eq. is got by first exchanging x and y in the Eq. 
above it, which amounts to projecting on 0Y\\ to OX, and then 

exchanging the letters in the angles, it being remembered that 

/— ^ /'-> ^-> ^—\ 

ab = — ha and sin ab=^ — sin ha. ^ 

Angles are best reckoned from the + X-axis or toiuard the 
-f- Y-axis. 

If the X'- resp. Y'-axis be sloped a resp. (3 to the X-axis, 
we may write 

X sin (xi = x' sin(a> — a)-\-y' sin(co — jS) , 

y sin (o = x' sin a -f- ?/' sin ^S. 

From these general formulae the student may find special ones : 

(a) For passing from rectangular to oblique axes. 

(b) For passing from oblique to rectangular axes. 

(c) For passing from rectangular to rectangular axes. 

The results are not so symmetric and easy to recall as the 
general formulae. Let the student draw the figures and inter- 
pret geometrically each term in each Eq. 

III. To 2^(^ss to parallel axes through a neio origin. Be OX, 
Fthe old axes, O'X', 0' F' the new ones ; x' , y', the old Cds. of 

the new origin 0'. If x, y 
resp. cc', y' be the old resp. 
new Cds. of P, we have 
x = x' + x^, y = y'-^y^, 

i.e., for the old Cds. put the 
neiu Cds. plus the old Cds. 
of the netu origin. 

If we will change both 
origin and axial directions, 
we can change first either, 
then the other, or both at 
once, by adding to the ex- 
pressions for the old Cds. the old Cds. of the new origin. 
Calling these latter, as above, x-^, y^, and putting q^, gg? Qi-, Q-i 




X 



LINEAR SUBSTITUTION. 25 

for the sine-quotients in II., we get as the most general rehitions 
between the Ccls. of a point referred to two systems : 

x = x^^ gi«'+ q^y'^ y = yi + gi'.^'+ g2'3/- 

Conversely, such a pair of Eqs. may alwaj's be interpreted as 
a transformation of Cds. For x^, 2/1 ^^J ^^ taken as old Cds. 
of a new origin (or negative new Cds. of an old origin), and 
we can find w, a, and /5 such that sin(co — a) : sina> = gi, 
sin (co — /5) : sin w = q^, etc. 

22. Note that the general Eqs. of transformation are homo- 
geneous of 1st degree in Cds. So much might have been 
assumed, it being clear tliat a length, as x or y, can be ex- 
pressed only as made up of lengths. Under this assumption, 
by determining the values of x^, i/^, and the g's, ttie student may 
now get the Eqs. already found ; this is recommended as a use- 
ful exercise. 

The magnitudes cci, 3/1, and the g's, are not of the same 
class ; the latter are pure numbers, trigonometric ratios, while 
the former are Cds., metric numbers of tracts. A number is 
said to have as man^- dimensions as the geometric magnitude it 
stands for : the metric number of a lengthy area^ resp. volume 
has one, tivo resp. three dimensions. A pure number, the ratio 
of two like metric numbers, is of 0th degree or dimension. 
Thus, 64^ = 8- = 4" has one, two, three dimensions, according as 
it is the metric number of a length, an area, a volume. 

It is plain that any Eq. may be thought as homogeneous by 
thinking the numeral coefficients of proper dimensions. 

23. If in any Eq. /(.^•, y) = 0, we put for x and y any 
linear functions (i.e., functions of 1st degree) of x' and ?/', we 
are said to make a linear substitution or transformation. Such 
a substitution may, of course, change the form of the Eq., but 
it ivill not change its degree in x and y. For it cannot raise 
the degree, since any term or factor of a term, as x% will be 
replaced by a series of terms, none of degree higher than the 



26 CO-ORDINATE GEOMETRY. 

9-th, in x^ and ?/' ; neither can it lower it, since then by express- 
ing a;' and ?/' as clearl}^ as we can, linearly through x and y^ and 
re-substituting, we should get the original Eq., and so raise the 
degree by a linear substitution, which is impossible. 

This again we might have foreseen. For the picture of 
/ {x, y)= is a curve whose degree tells the number of 
points in which it may be cut by a RL. (see Arts. 24, 16) ; a 
linear substitution is interpreted geometrically as a change of 
axes ; by such a change we in no wise affect the curve, hence 
do not change the number of points in which a RL. cuts it ; 
hence we do not change the degree of the Eq. 

The doctrine of Transformation of Cds. is of special impor- 
tance to Mechanics. Any motion of a plane system of points 
may be resolved into n push and a turn. A j^?6s/i corresponds 
to a change of origin simply, a turn to a change of axial 
directions; a tivist corresponds to a change of both. 

Illustrations. 1. Transform x' -\- lAx -{- y^ — 10 y -{- 4^9 = Q. 
to II axes through ( — 7,5). 

We have 

x = x'—7, y = y'-{-5; 

.•.(x'-7y+U(x'-l) + (y-j-5y-10{y'-i-D)-\-4.d = 0; 
or, x" + y" = 2D. 

2. X? — y^ — a^. Pass to axes halving the A between the old 
ones. 

We have 

(0 = 90°, a = 45°, /?=135°; 



.-. CO - a = 45°, w - ^ = - 45° ; 
.'.x = x' sin 4:5°— y' sin 45 



" V2 ' 



y = x' sin 45° + y' sin 135° = — ^^ ; 

V2 
whence, substituting and dropping accents, 

2xy = — or. 



LINEAR SUBSTITUTION. 27 

3. Sx^+4.xy-{- 5y^- 5x'-7ij-21 = = f(x,y). Change 
the origin and turn the axes, keeping them rectangular, so as to 
make the terms containing the first powers and the product of 
X and y vanish. 

Putting x'-\- Xi for x, y'-\-yi for y, and collecting, we get 

3a;'2 4. 4,x'y'+ 5?/'2 + 2{3x, + 2y^ ~i)^' 

+ 2(20^1 + 02/1 - i)y'-hf{x,,yO = 0. 

If the terms in x' and y' vanfsh, then 

3a^i + 2^/1- 1 = 0, 2a;i + 5?/i-|-=0; 
whence, a.'i = 1 : 2, yi= 1 : 2. 

Hence, /(x'l, 2/O =/a,i) = 3-^+ 4 .1 + 5 -i- 74 
-54-21 = -24. 

Accordingly, on passing to new U axes, through (^,^), the 
Eq. becomes 

Sx'^ + 4:x'y' -\-5y"--2i = 0. 

Now turn the axes through an ^ a ; then, X and Y being 

new axes, 

x'= X cos a — y sin a, y'= X sin a -{-y cos a ; 



.*. (3 cos a" + 5 sin a + 4 sin a • cos a)a;- 



+ (3 sin a 4- 5 cos a" — 4 sin a • cos a)y^ 



4- (4 sin a • cos a + 4 cos a' — 4 sma)xy=24:. 
If the term in xy vanishes, then 



sin a • cos a + COS a" — sin a" = 0, 
or, ^ sin 2a = — COS 2a, or, tan 2a = — 2. 

Hence, on reduction, 

(4 +V5)a^2^(4-V5)2/'=24, 

a is 58° 16' 57" or -J 31° 43' 3"^ ; but it is needless to find a 
from the tables ; it is much better to construct it geometrically. 



28 CO-ORDINATE GEOMETRY. 

We pass now to the geometric interpretation of P^qs., and 
naturally begin with the general 

Equation of First Degree in oc and y. 

24. This has the form Ix + 'niy -\-n= 0. 

To interpret it, let us pass to a new system of Cds. x' and y', 
such that x' = lx -{-my -]-n. Every point whose old Cds., 
X and ?/, satisfy Ix -}- my -f ?i = has its neiv Cd. x' = 0, 
and clearly all such points are on the F'-axis ; again, every 
point on the I^'-axis has its x'= 0, and hence has such old 
Cds., X and y, as satisfy lx-\-my -\-n = \ this Y'-axis is a 
RL. ; hence every point whose rectilinear Cds., x and ?/, satisfy 
an Eq. of 1st degree in x and y lies on a certain RL., and the 
Cds., X and y, of every point on that RL. satisfy that Eq. 

Conversely, suppose given any RL. , and seek the form of its 
Eq. Assume it as a new Y'-axis ; for all points on it and for 
no others ic' = ; but any Cd., x' , is a linear function of 
the old Cds., x and y ; i.e., x' = Ix + my + n ; hence for all 
points on this RL. and for no others lx + m.y-^n—0. 
Therefore, 

I. The geometric picture of any Eq. of 1st degree in rectilinear 
Cds., X and y, is a RL. 

11. The cdgehraic p)icture of any RL. is an Eq. of 1st degree in 

rectilinear Cds., x and y. 

We speak indifferently of the Eq. and of the RL. : 
Ix + my + n = 0. A convenient abbreviation for Ix + my + n 
is jL ; if I, m, n, have any subscript, L has the same subscript, 
and conversely, so that L,, = 4^ + m,,?/ + 7i^. We shall also 
speak of tjie right line L, meaning the RL. whose Eq. is 
L = 0. 

The above proof is the most natural, and presents no diffi- 
culty ; but owing to the great importance of the proposition, it 
will be well for the student to frame a proof fj-om the figure : 
showing that the Eqs. of the 4 RLs. are really such as are 



EQUATION OF THE EIGHT LINE. 29 

written by them ; thus showing that by changing a, &, s one of 
these Eqs. may be made to fit any EL. that may be drawn in 
the plane. 




25. The vahies of x and y range in pairs picturing the 
points of the RL. each from — oo to + «^ ; hence x and y are 
called running or current Cds. — For any one RL. Z, m, n 
are not definite, since we may multiply the Eq. by any expres- 
sion we will, without changing the relation between x and y. 
But the ratios l:m, m:n, nil are Jixecl for any one RL., 
different for different RLs. They are called arbitraries or 
parameters. Their number is apparently three., really two., for 
the tliird is but the quotient of the other two. Clearly they are 
not changed by multiplying the Eq. at will. To interpret them, 
assume any axes, and construct the RL. To do this it suffices 
to know tivo points of the RL. or one point and the direction. 
To find a point, we must find a pair of values of x and y satis- 
fying the Eq. To do this, we may assign an}^ value, say, to a?, 
and reckon the corresponding value of y. The simplest value 
we can assign to i» is ; the corresponding point will be on the 
Y'-axis, since only on the y-axis are points whose a? is 0. The 
corresponding value of ?/ is — n: m. This, then, is the distance 
Oly from the origin at which the RL. cuts the Y'-axis. Likewise, 
putting y = 0, we find the distance from the origin at which the 
RL. cuts the X-axis to be —n: I. The RL. making these inter- 



30 CO-ORDINATE GEOMETRY. 

cepts, —mm resp. —n:l on the Y- resp. X-axis, is the 
RL. Ix + my -{-n==0. 

Two of the ratios are thus seen to represent negative inter- 
cepts made on the axes by the RL. Denote these intercepts 
on the X- resp. y-axis hy a resp. 5, so that a=— ri:Z, 
&= —mm. Then, on transposing n and dividing by it, the 

Eq. becomes — I — =1, which is the Intercept Form (I.F.). 
a b ^ ^ 




The 3d ratio —l:m= let us denote by the symbol s 

and name the Direction-Coefficient of the RL. Note that s is 
the negative ratio of the coefficients of x and y. If be the 
slope of the RL. to the X-axis, then clearly 

s = sin $ : sin ((d — $). 

Hence for (o = 90°, i.e., for rectang. axes, 8=tan 0. 
Solving the Intercept Form as to y, we get y = SQC-{-b, 

which is the Directional Form (D.F.). 

Drop from on lx-\- my + 7i = 0, a J_ p sloped a resp. ^ 
to the X- resp. T-axis ; then p = a cos a = 5 cos /5. 

Substituting in the I.F. for a and &, we get 
X cos a + 2/ cos /5 — i> = 0, 
which is the Normal Form (N.F.). 



THE NORMAL FORM. 31 

For (1) = 90°, cos^=:sina; hence, we get 
i» cos a + 2/ sin a — j> = 0, 
an important special form. 

Hence we have the following Rules : 

To bring the Eq. of a RL. to the I.F., divide by the absolute 
term taken to the right member. This is possible unless the 
absolute be 0; then the pair (0,0) satisfies the Eq., and the 
RL. goes through the origin. — In general, if the absolute in any 
Eq.^ of any degree^ betiueen Cds. be 0, the curve goes through the 
origin. — To construct a RL. through the origin, assign either 
Cd. any convenient value, then reckon the other Cd. ; this pair 
pictures a point, which with the origin fixes the RL. 

To bring to the D.F., solve as to y. This is possible unless 
the coefficient of y be 0, i.e., unless y does not appear, at all; 
then the Eq. reduces to a? = a constant, — the RL. is || to the 
Y-axis. Likewise, if x does not appear, y = Si constant, — 
the RL. is H to the X-axis. 

To bring to the JSf.F., multiply by the normalizing factor F. 

To find F., we note that, since by hypothesis 

Fix -\- Fmy -f Fn = x cos a-j-y cos (3 —p, 

Fl = cos a, Fm = cos y8, Fn = — p, 

whence, F— —p : n. 

j ( 7i^ , n^ ^n n ") n n . 

i\n 7^ + — -2-. -cos (o \ =_._sm(o, 

since each is the double area of the A 01 Jy. Hence 



F= sin CO : V^^ + m' - 2 Im cos co. 

Before the y' we have choice of signs ; we agree to take 
always that sign which will make the absolute negative. 

For (0 = 90°, the most important case, 
2^=1 : ■yJP-\-m?, 



32 



CO-OEDINATE GEOMETEY. 



EXERCISES. 

Construct the following RLs., reduce the Eqs. to the various forms, for 
oj = 90^, unless otherwise stated : 
1. 5a; -3?/ +30 = 0. 

The I.F. is 1- -^ = 1, got by dividing by — 30; 

— 6 10 

the D.E. is v/ = |a: + 10, got by solving as to ?/ ; 

5 . 3 



the N.F. is ^^— x ■\ ;:^ = 0, got by multiplying by F= —1 : v34. 

\/84 \/34 

a=-Q, 6 = 10, s -= 5 : 3 = tan 0, p = 30 : V34. 

For co= 60°, a, h, s are the same, but F= — \/3 : 14, p =15\/3 : 7. 

2. 3r+7 = 0. 4. 3:r-2?/=0. 6. 2:r +4?/ = 9 (w = 60^). 

3. 7?/ -9=0. 5. 3a: -4?/ = 12. 

7. How are a: + y = and x — y/ = 0, a; — «/-}-a = and x -\- y +6=0 
related (oj at will) ? 




26. Angle betiueen two E.Ls. : Zii:c + mi?/-f-72i = and 
l^x -{- m2y -i- n2 = . The ^ (^ between the RLs. equals the 
^ ai — ttg between the J_s j9i, p2 let fall on them from the origin. 

By Art. 25, 

cos tti =: li sin CO : V^i^ + m^^ — ^l-^rn-^ cos co, 



cos a^ = Z2 sill CO : -y/l^ + m2^ — 2 ^2^2 cos co. 



ANGLE BETWEEN TWO RIGHT LINES. 33 

Square, take from 1, extract square roots ; there results, 



sm 



tti = my— li COS cj : -Vli + 'i^ii —2limi cos 



sin a2 = W2 — 12 cos 00 : V(2). 

From applying the Addition-Theorems of Sine and Cosine 
there results : 

sin<^ = sin (aj— 03) = (?im2 — Z2mi)sin oa : ■\/(l)- V(2) ; 

COS^ = COS(ai— ttg) 

= lhh-\-'^i ''"2 — h '^^h + 4 ^1 COS CO I : V(l)-V(2) ; 

tan ^ = tan (ai — a2) 

= (Zim,— Z2mi)sinw : ^ZiZ2-|-miW2— ^imgH-^^^iCOSw^. 

Special Cases. 1. If the RLs. be || , <^ = 0, .•.tan<^ = 0, 
. • . liiUo — lo'Tiii = ; or, li : m^ = ?2 = "^2 ; c>i'5 ^1 = ^2 ; i.e. , 

MLs. are || tvhen, and only ivhen^ their Direction- Coefficients 
are equal. 

2. If the RLs. are J_, <^ = 90°, .'. tan (/) = go , 

. • . ?! Z2 + 'i^h '^'>h — (^1 W2 + 4 '^i) COS to = 0. For o) = 90°, 

iiZ2 + ^i'^2= 0, or -^ = =, or Sj = ; i.e., 

ul-^ C9 02 

In rectangular Cds. two RLs. are ± ivlien., and only ivlien., 
their Directioii- Coefficients are negative reciprocals of each other. 
This is the case when the coefficients of x and y in the one Eq. 
are the exchanged or inverted coefficients of x and y in the 
other, with the sign of one of them changed. 

The absolute term affects neither perpendicularit}' nor paral- 
lelism. The rectang. Eq. of the RL. through {x^., 2/1) -L to 
Ix -\- my + 71 = is I (y —y^) = m' {x — Xi) . Why ? 

Illustrations. 1. The sides of a A are : ?> x — 'iy -{- 12 = ^ 
5x+ 2?/ + 10 = 0, a; + 52/ + 5 = ; find its angles (co = 60°). 



34 CO-ORDINATE GEOMETRY. 

tanc^,= (25-2). ^V3 ^ j3 
5 -f: 10 -(25 + 2). i V3 

whence <^i = 85° 47' 28". 

Find (ji2 and </)3, check by (^i + </)2 + <^3 = 180°, and construct 
the A. 

2. Fmd Eq. of RL. through (1,2) ± to 3x + 5y=^7, and 
draw it, for <o = 90° and to = 60°. 

27. Distance from a point {x',y') to the RL, 

XGOSa-{-y COS /3 —p== 0. 
The Eq. of a RL. |I to the given RL. is 

XGOSa-{-yCOS^—p' — 0. 

If this RL. goes through (a?',?/'), then 

x' cos a + 2/' cos 13 =p'. 
Subtracting j> =p, we get 

x' cos a -j- y' cos /3 — p = p' — p, 
which is clearly the distance sought. 

Y 




This result is + or ■— according as {x',y') lies on the outer 
or inner side of the RL. (the inner side being next to the 
origin) . Hence the Rule : Reduce the Eq. to the jSf.F. , p?<^ for 
the curreiit Cds. the Cds. of the point ; the result is the metric 



THE EIGHT LIl!^E UNDER CONDITIONS. 35 

number of the distance from the point to the RL., and is + or 
— according as the point lies on the outer or inner side of the 
RL. If we put ^= ce cosa + 2/ coS;8 — p, then N" is the 
distance of (x, y) from N = 0. ^changes sign, passing through 
0, as {x^y) changes sides, passing through N^O. 

Carefully distinguish between the expression N and the Eq. 
JV= 0. In N^ X and y are Cds. of any point in the plane ; in 
N= 0, X and y are Cds. of any point on the RL. N= 0. It is 
this double use, bewilder though it ma}^ at first, that gives the 
N.F. its importance. 

Illustrations. 1. Find the distance from (3, —4) to 

Ax + 2y = 7. 

For o) = 90°, N^F. is ^ + -^ ^ = ; hence, the dis- 
tance is -3 : 2V5. ^''^ V5 2V5 

For u)= 60°, the distance is —J. The point is on the inner 
side of the RL. 

2. Find the distance from (2, 3) to 2a? + ?/ = 4. 

3. Find the distance from the origin to a{x—a) -{-b(y—b) = 0. 

The E-ig-ht Line under Conditions. 

28. By Art. 25 the Eq. of the RL. contains two arbitraries 
or parameters. If we hold one of these fast, for every value of 
the other we get a RL., and for the totality of values from — oo to 
+ 00 we get a family or system of RLs. Thus, in y = 8x-\-b, 
holding b fast and assigning s the whole series of real values 
from —00 to 4-00 5 we get a family of RLs., all cutting the 
y-axis b from the origin. Loosing 6, assigning it the same 
series of values, we get a family of families, one through every 
point of the y-axis. Except this Y-axis, which is common to 
all the families, no RL. of one family is a RL. of another : all 
are different RLs. ; i.e., all possible different RLs. in a plane, 
enough to Jill the plane, form afaynily of families, an infinity of 
infinities of RLs. ; i.e., the plane viewed as full of RLs. is doubly 



36 CO-ORDINATE GEOMETRY. 

extended. Hence, to know a RL., we must know two things 
about it : what member of what family it is ; and to fix a RLo 
we must put it under two conditions : we must determine the 
two parameters in its Eq. We consider here some simplest 
cases. 




I. To find the Eq. of a family of RLs. through a point 

The Eq. of any EL. is y = sx-\-h^ and the Eq. 2/1 = sx^+h 
says that the RL. passes through (o^i, 2/1) • This last is not an 
Eq. of a line, since it contains no current Cds., but an Eq. of 
condition. By its help we can eliminate s or &, better 6, and 
get 

y — yi = s{x — x^. 

For any one value of s this is the Eq. of a RL. (since it con- 
tains current Cds. in 1st degree only) through (i»i,.Vi), since 
that pair of values satisfies it. To any slope of such a RL. 
there corresponds a value of s; viz., s= sin^ : sin to — ^, and 
conversely; hence, letting s range from —00 to +00 , we get 
all RLs., the family of RLs., through (a.\,z/i). 

To determine a member of this family, we may impose various 
conditions ; as, that it go through (x22/2)- Hence, 



THE EIGHT LINE UNDEK CONDITIONS. 



37 



II. To find the Eq. of a RL. through tioo points. 

Since it goes through both (i»7i, 2/1) and {X2,y2)-, it is com- 
mon to the families 

y — y-^=z six — Xi) and y — y2 = s(x — X2) , 

and for it s has the same value in the two Eqs. ; hence, elimi- 
nating s, we get the Eq. sought: 

y — y-] '. y — y2 — x — x-^ i x ■^~ X2' 

Let the student interpret this proportion geometrically. 

We may reason otherwise ; thus : 

Be Ix + my -{-n = the Eq. of the RL. ; since it goes through 

(^1,2/1)5 

Ixi -\- my I + n = ; 

and, for like reason, 

1x2 + m?/2 -f- ^ = 0. 

By Introduction, Art. 14, these Eqs. consist when, and only 
when, 

0. . 



X 


y 


1 


x^ 


Vi 


1 


tJOt) 


2/2 


1 



This Eq. of the RL. is equivalent to the other, and very con- 
venient. 



Corollary. — 'Jl 



Xo 



or 



X, 


Vi 


1 


tAJQ 


2/2 


1 


X^ 


2/3 


1 







2/3 2/2 ^3 ^2 

is the Eq. of condition that (cc^, y^) , (X2, 2/2) 5 (%? 2/3) li® on a RL. 

III. To find the Eq. of a RL. through a given point (£»?i,2/i) 

and sloped cf) to a RL. ivhose Direction-coefficient is Si (co = 90°) . 

The RL. is of the family 'y — yj^ = s(x — Xi) ; also, by Art. 26, 



tan cfi 



I mi — li7)i 
lli-\-mmi 

•'• 2/ -2/1 



_ "^i 



s Si — tan ch 
— , or s = — ^ ; 

1 + ssi 1+^1 tan <jf) 



tan 4> 



1 -{- Si tan cji 



\ """ 1 / ' 



38 CO-ORDIKATE GEOMETRY. 

Corollary. If <^ = 90°, i.e., if the RL. is to be J_ to the 
given RL., 

2/ — ?/i = (x — Xi) or /i(y — ?/i) = mi(x — x^) . 

Si 

Let the student solve this problem for w not = 90°. 



EXERCISES. 

1. The vertices of a A are (5,-7), (1, 11), (—4, 13) ; find the Eqs. of: 
its sides, its medials, ±s through the mid-points of its sides, ±s from its 
vertices on the counter sides ; find the lengths of the last ±s. 



2. Find the Eq. of the EL. through {x^,i/^) cutting P1P2 in the ratio 

3. Eind the EL. through (5, 4) forming with + axes a A of area 80. 
(a) = 90°). 

4. The sides of a 4-side, taken in order, are y —- bx, Qy -\-bx — 35, 
Zx —y —21, 4?/+9x=0; find the Cds. of its vertices, Eqs. of its diag- 
onals, and of the junction-lines of their mid-points. 

5. Eind Eqs. of ELs. through the junction-point of 3a: — 4?/ = 7 and 

2x + 5?/-|-8 = 0, and sloped 60° to ?/ = 4 x -f 3. 

6. Eind Eqs. of ELs. sloped 30° to X-axis and cutting Y-axis 7.5 
from origin. 

7. Are (6,2), (7,-3), (-5,-5), on a EL.? Are (3,-1), (1,2), 

(7,-7)? 

8. Two counter sides of a 4-side being axes, the other sides are 

— — -f -^= 1, -f- -^= 1 ; find the mid-points of the diagonals. 

2 a' 2h 2 a' 2 6 

29. The parameter of a family of RLs. is the arbitrary in 
its Eq. 

Theorem. When the parameter appears in 1st degree only, 
all the RLs. of the family go through a point. 

Solve the Eq. as to the parameter A, we get 

A = (/ix -f m^y 4- n^) : {IjX -f m^y + n^) , 

or ?icc + mi?/ + 711 — -^ (''2^ + "^n^y + n^ = 0, 

or Li — XL2 = 0. 



PENCILS OF EIGHT LINES. 39 

Whatever X be, this RL. goes through the junction-point of 
the RLs. Li = and iy^ = 0, since the pair {x,y) which satis- 
fies both these Eqs. also satisfies Li — XL2=0. 

Conversely, all RLs. through the junction-point of L^=^0 and 
Xo = are of the family Li — kL.2= 0. For the Direction-coeffi- 
cient is s = (Xl2 — li) : {nil — X 7112) , ^^^ this ranges with A. 
through all real values. 

Hence, to find the Eq. of any special E-L. through the junc- 
tion of Li = and L2 = 0, which let us call the point {L^.L^)-, 
it suffices to find the corresponding special value of A, which 
we may call the parameter of that RL. Thus, if the RL. is to 
pass through the origin, then 7ii — Xng = 0, A = 7^1 : 719 ; if it is 
to be 11 to the X- resp. y-axis, then ?j^ — A/2 = resp. 
Wj — Am2 = 0, X = li:l2 resp. A = 7?ii : mo. 

The student may substitute these values of A, and find the 
Eqs. of the RLs. 

30. The above is a special case of the important theorem : 
the curve C^ — XC2 = goes through all junction-points of 
the curves Ci = 0, C2 — O (A being any constant). For 
any pair {x,y) that satisfies both Ci = and (72 = 0, sat- 
isfies also (7i — A(72 = 0. 

This again is a special case of the still higher theorem : If, 
when two Eqs. are satisfied, a third is also satisfied, the third 
curve passes through all junction-points of the other two. 

The proposition is evident as soon as its terms are under- 
stood. 

Hence, it is plain that the RL. /x^ii -f-f<-2-Z^2 = goes 
through the junction of Li = and L2 = 0, since its Eq. 
is satisfied when the others are. 

If the expression /^iiyiH-/.'oX2 + /^3-^3^ 0, i.e., vanish 
identically, then the three RLs. L^ = 0, X2 = 0, Ls= 0, pass 
through a point, since the pair that satisfies two of the Eqs. 
must satisfy the third. Hence, if the sum of some multiples of 
the Eqs. of three RLs., vanish identically, the three RLs. go 
through a point. 



40 CO-OPvDIKATE GEOMETKY. 

If fJ^iLi + /x2-^2 + /^3^3 = 0, but not identicalh', then this 
Eq. imposes some condition on the symbols x and y ; hence it 
is the Eq. of some line, and in fact of some RL., since x and y 
enter the Eq. in first degree only. We may go further, and 
say that this Eq. may be made the Eq. of any RL. by choosing 
the /x's properl^^ For Ix + 77iy -f- ?i = is any RL., and we 
may choose the three yu's so as to satisfy the three Eqs. : 

/Xili -T'fl2^2 ~rf-3^3 ^^^5 

/^i^h -{' 1^2'^h -\~ f^s'^h = ■^« 

The RLs. Li= 0, L2 = 0, ig = 0, determine a A, which 
may be called the A of reference^ or referee A ; and the expres- 
sions ivi, io, Z/o, may be called the trUinear, or triangular Cds. 
of points on the RL. fj-iLi + /x2-i^2 + /^s-^s == 0. 

If Fi, ¥2^ Fn^ be the normalizing factors of Li, L2, L., then 
FiLi^ FoLo, FqLq^ are the distances of any point (x,y) from 
the RLs. Zyi = 0, L2 = 0, ^3=0. Hence, it is seen that 
the triangulai" Cds. of a point {x,y) are certain fixed midtiples 
of its distances from the sides of the referee A. Triangular Cds. 
will be simplest, then, when they are the simplest multiples of 
the distances from the sides of the A ; i.e., when they are the 
distances themselves ; and these they are when, and only when, 
the Eqs. of the sides of the A are in the N.F. ; m the N.F. we 
write them, JV^ = 0, ^^2 = 0, ^3 = 0, where 

Hence, calling now the multipliers of the -^"'s v's, we have 

as the normal Eq. of a RL. in triangidctr Cds. : JVi, iVg? -^s- 

31. Here the point has tho^ee Cds. , the iV's, while we know that 
two are enough to fix it. But, since the Eq. is homogeneous in 
iV^'s, we can at once reduce the number of Cds. to two by 
dividing through by one, as iVg, and treating the quotients JSfi : ^3, 



TRIANGULAR CO-OKDIXATES. 



41 



N2 : N^ as the independent Cds. In general, the tnangular Cds. 
of a point are hound together hy a certain constant relation. For 
if Tve suppose, as we may, that the origin is within the 
referee A, and put ti, t2, rg for the tracts between the vertices, A 
for its area, then for ever}' point {x^y) or (^^1, N^-, N^ we shall 

This is clear at once when P is within the A, and is equally 
clear, on proper regard of signs, when P is without. By multi- 
plying this Eq. appropriatel}^ we can express any constant 
homogeneously through the triangular Cds. of any point. If 
the origin be without the A, it suffices to change the sign of 
one of the /Vs. 




A general test of whether three ELs. Xi = 0, i2 = 0, 
Xg = 0, go through a point, is found in the Determinant 



h 


mi 


7il 


l> 


Vlo 


V2 


h 


7^3 


^>3 



which, by Introduction, Art. 14, must vanish when the three 

^ * Ziic + mi2/ + % = 0, 

^2^ + m.jy -f- th = 0, 
and I3X -{- m^y -f- no = 0, 

consist, are all satisfied by the same pair {x,y). 



42 



CO-ORDINATE GEOMETRY. 



So too must 



T/j V2 V3 

7/j V2 V3 



= 0, 



when through a point go the RLs. 



Geometric Interpretation of X. 

32. A family of RLs. through a point we may call a pencil, 
the point itself, the centre of the pencil ; the two RLs. through 
which the others are expressed, the base-lines of the pencil. If 
the Eqs. of the base-lines be in the N.F., then iVi — AiVo = 0, 
for any special value of A, is the Eq. of a RL. of the pencil. 



^ 


1 




^ 


c 

II 
1 








A 







1 


-f 1 





Here X = i\^i : ^2 = ^«^*'o of the distances of any point of the RL. 
N^ — XN,_ = from the base-lmes JVi = and iV^s = ; or, X = 
ratio of the sines of the slopes of the RL. N-^ — XN^^O to the 
hase-Unes N^^O and A^2 = 0- 

If we call the angle containing the origin, and its vertical 
angle, and the hues in them, all inner, the others all outer, then 
we see that for inner lines the distances are like-signed, as 
from Pi, P3, and .'.X is + ; for outer lines the distances are 
uiilike-signed, as from P2, P41 ^^^ .•. X is — . 



ABRIDGED XOTATION. 43 

If we call the inner side of a RL. also the — side, the outer 
also the + side, then angles and sines of angles reckoned from 
the + resp. — side of a RL. are themselves + resp. — ; we 
agree to reckon angles from the fixed to the variable RL. 

33. For the inner resp. outer halvers of the angles at (jVi, N2) 
the distances are equal and like- resp. unlike-signed ; hence, 
X= +1 resp. —1 ; hence, the halvers arie N^ — N2 = resp. 

Hence, to find the Eq. of the inner resp. outer halver of the 
^s between two RLs., form the difference resp. sum of their 
Eqs. in the N.F. 

N.B. Of the two equivalent Eqs., ± ^= 0, we have taken 
that as the N.F. in which the absolute term is negative. But 
this test fails, and with it the test of which is the + and which 
the — side of the RL., when the absolute term is 0, i.e., when 
the RL. goes through the origin. In this case, we may agree 
to take always the term in ?/, or alwa3's that in x^ as positive. 
If we make the first agreement, as is common, the + side will 
lie next to the + y-axis ; for, holding x and letting y increase, 
we must get a + result in the left member of the Eq. 

34. This Abridged Notation (a single letter N standing for 
the left member of the Eq. of a RL. in the N.F.) with its imme- 
diate outgrowth, the system of triangular Cds., yields a method 
of great strength and beauty ; we have space for but a few 
simple 

Illustrations. 1. The inner halvers of the ^s of a A meet 
in a point. For, be the origin within the A, and ^^ = 0, 
N2 = (i-, -A^3 = 0, its sides, then the sum of the Eqs. of the 
inner halvers, Ni — No^O, N2 — N^ = 0^ ^3 — -^i = 0, van- 
ishes identically. 

2. Two outer halvers and the third inner halver meet in a 
point. They are iV2 + ^3 = 0, N^-]-N^ = Q, ^i-iV, = 0; 
multiply the second by —1, and sum. 



u 



CO-OEDIi^ATE GEOMETRY. 



3. The _Ls from the vertices of a A on the counter sides 
meet in a point. The ^s being Ai, ^21 ^3? the _L from Ai on 
iVj = is N'2 cos A2 — iVs cos ^3=0; permute the indices, and 
sum. 

4. If through each vertex of a A be drawn a pair of rays 
hke-sloped to the sides meeting there, the junction-lines of the 
junction-point of each couple next to a side and the counter 
vertex meet in a point. 




Be -ZVi = 0, iV2 = 0, JSI's = the sides, A^, A2, A^ the 
counter vertices, yi, 725 ys the slopes of the pairs at the ver- 
tices. The Eqs. of A2A1 resp. A^A^' are 

JSf^ sin (ys + A) + ^3 sin ya = 0, 

resp. Ni sin (yg + A^) + JV2 sin yg = 0. 

The Eq. of A^Ai' is, therefore, 

Wi sin(y2 + A2) -\- iVg sin yg — X^JSf^ sin yg + ^3 -f- N'2 sin ys^ = 0. 

At Ai both W2 and JV3 are ; hence. 



A = sin ys -f ^2 : sin y3 -f A^. 



.-. ^3 • sin y2 • sin y3 + ^3 — JV2 • sin y3 • sin yg + ^2 = 
is the Eq. of AiAi' ; 



JSfi ' sin y3 • sin y^ -f- ^^ — iV"3 • sin yi • sin yg + ^3 = 
is the Eq. of A A' J 



POLAR EQUATION OF THE EIGHT LINE. 45 



N2 • sin yi • sin yo + ^2 — -^i • sin yg * sin yi + ^1 = 

is the Eq. of A^A^'. 

Multiply by sinyi, sinya resp. sinyg, and sum. 

If A3A1' meets A^A^,' in ^2"? show that ^2^2"? A^A^", AA" 
meet in a point. 

5. If through each vertex of a A be drawn a pair of rays 
like-sloped to the halver of the ^ at that vertex, and if any 
three of these rays meet in a point, so will the other three. 

If 2^1 sin ^2 +72— -^3 sin yg = 0, -^9 sin^^g+yg— -^1 sinyg = 0, 
^3 sin Ai-{-yi — N2 sin y^ = 0, be anj- three, the others are got 
by simply exchanging the JV's in each Eq. The value of any 
N in one of these Eqs. is not in general the same as its value 
in another ; but if the first three RLs. meet in a point, the first 
three Eqs. must consist for one set of values of the JVs ; hence 
we may transpose, multiply, cancel the product N1N2N2,', and 
get as a condition that the three RLs. meet in a point, the Eq., 



sin ^2 + 72 • sin ^3 + yg • sin ^^ + y^ = sin y^ • sin yg • sin yi. 

Now, clearly, exchanging the ^'s in each Eq. affects not this 
result. 

6. If the three _Ls from the three vertices of one A on the 
three sides of a second A meet in a point, so do the three _Ls 
from the three vertices of the second on the three sides of the 
first. 

Be the sides of the one A iVi = 0/ ^2 = 0, JVg = 0, the sides 
of the other iVi'= 0, ^^' = 0, iVg'^O; then 

iVs cos jffjS's = TV;,. cos jffj^o, iVs'cos i^g'= JVg'- cos iQt^' 

are a pair of corresponding _Ls. Let the student complete the 
proof. 

Polar Equation of the Right Line. 

35. If the _L from the origin on the RL. be sloped a to the 
polar axis OD, we have at once, 



46 



CO-OEDINATE GEOMETEY. 



pCOs(^— a) =p 

Let the student get this Eq. by transforming to polar Cds. 
from the N.F. 




EXERCISES. 



1. Reduce p=: 2a sec[ + — j to rectang. Cds. 

2. Find where p=asec|0— — ] and pcos[0 — -] = 2a meet, and 
under what angle. ^ ^ ^ ' 

3. Eind where the ± from the pole meets the RL. through {pi,0-i). 

Miscellany. 

36. The Eq. of the RL. through {x2^y^^ (^352/3)5 in the 
N.F. is 



X y 1 sinaj:Vl2/2— 2/3^+^2— ^3^+22/2— 2/3-^'2— ^'3- coscoj = 0. 
^2 2/2 1 
^3 2/3 1- 

The left side is the distance from {x^ y) to the RL., the •>/ is 

1 1 

the length of the tract (^21 2/2) (^35 2/3) 5 hence, their product, 

(^152/1)5 heing written for (x^y), or 

sin w, 



x^ 


2/1 


1 


X2 


2/2 


1 


X3 


2/3 


1 



is the double area of the A whose vertices are (a^i,2/i)? (^252/2) 
(^35 2/3) • 



AREAS. 



47 



37. To find the area of a polygon whose vertices (taken in 

order) are 

(rKi,?/i), {x,,yo) ••• {x^^Vn)' 

Note that when one vertex of a A is the origin (0,0), its 



double area is 



^'i 2/r 

X.2 2/o 



5 (^152/1)5 {'^■i^y-i) being the other vertices, 

the axes assumed rectangular. Now from the origin, assumed 
within the polygon, draw rays to the vertices cutting it up into 
n A ; the double area of the polj^gon then is 



Xi X2 + 


Juo *^^ 


+ • 


•• + 


^n-\ ^n 


+ 


X^ Xi 


IJl 2/2 


2/2 2/3 






2/n-l Vn 




Vn 2/1 



^zr 


1 2 


+ 


1 


+ 


Xq Xo 


+ 


it'y ^'q 




2/1' 2/2' 




2/1' 2/0 




2/« 2/2' 




2/0 2/0 



Now suppose the origin moved to (0.-0,2/0) I the double area 
of the first A changes to 

•Xyi "y" Jbf\ • tX/9 **j~ Uyf) 

2/1' +2/0, 2/2' +2/0 

The first of these Determinants is the original primed, the 
last vanishes, the sum of second and third is 

My 2- 2/1') + 2/0 («^i'- ^2) ' 

If we operate likewise upon the other Determinants, we shall 
clearly get the original Determinants primed, while' the sum of 
the multipliers of x^) and 2/0 "^i^l each vanish identically. Hence 
wherever the origin be, the double polygonal area is 



Qy-i tJut} 


+ 


X2 2/2 


+ • 


• + 


X, x^ 


2/1 2/2 




^3 2/3 






Vn 2/1 



the primes being dropped. 

We thus learn that the sum of the areas of A ivith one com- 
mon vertex^ the other vertices being vertices of a polygon^ is the 
area of that X)olygon : a theorem important in mechanics. The 
practical rule is to write the x's and y's in order, each in a hori- 
zontal row, repeating the first of each as the last ; thus, 



1 *^ 3 

2/1 2/2 2/3 



Vr. 2/1' 



48 CO-ORDINATE GEOMETRY. 

and take the cross-products of the consecutive paks with oppo- 
site signs. The algebraic fact that the sign of the area is + or 
— according as we take one or the other set of products + , 
corresponds to the geometric fact that we may compass the 
polygon clockwise or counter-clockivise. 

For axes not rectangular, multiply by sin to. 

38. To find the area of the A ichose sides are ij = 0, Xg = 0, 
iyg = 0, form the Determinant of the coefficients ; 



^1 


mi 


Ui 


4 


mg 


712 


4 


m^ 


Tig 



and denote the co-factor of any element, as Z^, by that symbol 
primed, li ', be (x^^^y^) the junction-point of L2 = 0, L^=0. 
Then we have i»i = //:%', yi = mi':ni, and so for {x^^y^^ 
(^31 2/3)- Form the Determinant of Art. 36 ; replace the column 
of I's by Ui'-.Tii'^ n^:n^^ nJ : Uq' ■ set out the divisors of the 
rows, %', na', nJ ; there results 

2 A =1 Zi'mg'ng'l sinw : n-f^-n^'n^ 

= I ZirngTisI^ sin w : | Zi??i2 1 • ^'^^s 1 • 4^i | • 

If two of the RLs. be 1| , a factor of the denominator van- 
ishes, the area is 00 ; if the three ELs. meet in a point, the 
numerator vanishes, the area is 0. 



39. To find the ratio /xj : /xg in which the tract (iCi, 2/1) (ccg, yo) 
is cut by the ML. Ix + my -{- n = 0. We have at once, on can- 
celling the normalizing factor i^, 

/xi Pi Ixi + riiyi + n 

When the section is inner, /xj and ^^ are like-signed, p^ and P2 
unlike-signed ; when the section is outer, the /x's are unlike- 
signed, the j9's like-signed ; hence the — signs in the above Eq. 



TRANSVERSALS. 



49 




X y 


1 


^3 Vz 


1 


^A 2/4 


1 



^1 Vi 1 


: 


X2 2/2 1 


% ys 1 




«^3 2/3 1 


x^ 2/4 1 




X4 2/4 1 



If the RL. go through (iCg, 2/3) and (iC4, 2/4) , its Eq. is 
= ; hence, />ti : /X2 = — 

P2P,P, 

40. By the help of these expressions for /xj : /xg we can now 
prove two fundamental Theorems on Transversals : 
I. Three points on a BL. and the sides of a A cut its sides into 
segments whose compound ratio is — 1 . 
II. Three RLs. through a point and the vertices of a A cut its 
sides into segments whose compound ratio is + 1 . 




Be ABC the A, L~lx-\- my -\-n = the transversal, cutting 
the sides at ^', 5', C" ; be j,L~lXj^-\- my;, -{-71 the result of 
putting Xj,, yj^ in L for x^ y. Then, by Art. 39, 



50 



CO-ORDINATE GEOMETRY. 



BA': A'C^-^L-.sL, 
CB' :B'A=-^L:,L, 
AC: C'B^-,L:,L; 

the product of these ratios is —1, which proves I. 

Be {x,y) the Cds. of P, through which are drawn AA", 

BB", CC". 

X2 1/2 1 
Xi ?/i 1 
X y 1 



Then 

BA" 

A"0' 





^3 2/3 1 


I 


Xi y^ 1 




X y 1 



^3 2/3 1 




^1 2/1 1 


2 2/2 


: 


^2 2/2 1 


a; ?/ 1 




X y 1 



X, ?/i 1 




X2 2/2 1 


^•3 2/3 1 


: 


^3 2/3 1 


a? ?/ 1 




X y 1 



B'A 



AC" 
C"B 



In the product of these ratios the determinants of the denom- 
inator are those of the numerator with two rows exchanged ; 
i.e., with signs changed; hence, the product is +1, which 
proves II. 

Note the order of letters and indices. 

EXERCISES. 

1. Find the areas of the A whose vertices are : (2, 1), (3, —2), (4, — 1); 
(2,3), (4,-5), (-3,-6). ^ns. 10; 29. 

2. Eind the area of the tetragon : (1, 1 ), (2, 3), (3, 3), (4, 1). 
12 3 4 1 



Solution. 



13 3 11' 3 + 6 + 3 + 4-2-9- 



12 - 1 = - 



Ans. 4. 



3. rind the area of the A : 2x — Sij = 5, 4:x+ 5i/ — — 7, y — 2x =9. 



Cross Ratios. 

41. A set of points on a RL. is called a range or roiv; a set 
of RLs. through a point, si pencil; each (half-) RL., a ray. 

The RL. is the carrier of the row or range ; the point, the 
centre or carrier of the pencil. 



CEOSS RATIOS. 51 

The distance-ratio of Pg to Pi and P3 is P1P2 : P2P3 (Art. 17) ; 
and so the distance-ratio of P4 to Pj and P3 is P1P4 : P4P3. 

The ratio 0/ the distance-ratios of two points of a range to two 
other points of that range is called the cross ratio of the range ^ 
and is written 

\P1P2P3P4l = ^^ : ^^'' 

Observing signs, we see at once, 

lPjP2P3PA = -— ''^'^^ ^ 
^ ' ' ' '^ P,P,.P,P, 

or, simply, Jl 2 3 4j — ' ? 

23.41 

which is the neatest way to write it. 

Clearly, the order of the points is essential. The alternates 
in position^ as 1st and 3d, 2d and 4th, are called conjugates; 
the consecutives are non-conjugates ; as 1st and 2d, 2d and 3d, 
3d and 4th, 4tli and 1st. An}^ one has one conjugate, two 
non-conjugates. 

The four symbols may be permuted in 4 !, == 24, ways. Any 
permutation or order may be got from an}^ other, as 1 2 3 4, by 
one or more exchanges, which will be either of a pair of conju- 
gates or of a pair of non-conjugates. 

To exchange a pair of conjugates inverts the ratio. 
Be jl 23 4;^ ij'ij = r; 



then Jl 4 3 2 



2 3-41 

r4-3~2 ¥^'IT 1 



43.21 12.34 r 



To exchange a pair of noji-conjugates takes the complement of 
the ratio to 1. For, exchange (say) 2 and 3 ; then 



52 CO-ORDINATE GEOMETRY. 



3 2.41 3 2. 41 



^ 12-3 4 2 3(1 2 + 2 3 + 3 4) 
3I.4T 3~2.0 



12-34,23.14 , 

+ z= — = = 1 — r. 



23-41 32-41 

Hence, to exchange two pairs of conjugates, or two pairs of 
non-conjugates, keeps the cross ratio unchanged. Hence there 
are four permutations for which the cross ratio (or CE-.) is the 
same ; hence, too, there are six permutations, or six sets of 
four permutations, for which the CRs. are different. By invert- 
ing and complementing to 1, we get these values : 

1 1 1 1 1 1 ^ 

r, ^, 1-r, , 1--, 1- 



r 1 — ?' r 1 —r 

The circle of these six A^alues is complete ; any amount of 
inverting and complementing will reproduce one of them. The 
last two may be written 

r — 1 T T 
and 



-1 

The ratio of the sines of the ^s into ivhicJi a third ray cats the 
^ between tiuo other rays is called the sine ratio of the third ray 
to the other tivo; the ratio of the sine ratios of two rays to two 
others is called the cross ratio of the rays^ and is written 

/~^ /'~^ /-^^ /-~^ 

cy< 1 00 ^ > sin 1 2 sin 1 4 sin 1 2 • sin 3 4 
/O ^ i z o 4^ = : = — , 

sin 2 3 sin 4 3 sin 2 3 • sin 4 1 

when jS is the centre of the pencil, and the ra^'s are 1, 2, 3, 4. 

When the rays of a pencil go through the points of a orange, 
the two are conjoined. 



CROSS RATIOS. 53 

From the figure, we see at once 

p-n = Sl'S4.-smfl, p'2~3 = jS2'S3'BmP3, 
whence, Jl 2 3 4J = aS'J1 2 3 4J ; or 

T7ie CRs. of a conjoined range and pencil are equal. 




Hence, holding the carrier of either range or pencil fast, we 
may move the other at will. 

If Jl 2 3 4^ = -l = /S"Jl 2 3 45, the tract between two 
conjugate points, resp. ^ between two conjugate rays, is cut 
innerly and outerly in the same ratio by the other conjugates ; 
the range resp. pencil is then called harmonic, and either j9a{r 
of conjugates (points or rays) is called harmonic to the other 
pair^ while the fourth element (point or ray) is called a fourth 
harmonic to the other three in order. 

If in an harmonic range one point halve innerly the tract 
betiveen two conjugates, the other (its conjugate) 7)iust halve the 
same tract outerly, i.e., must be at cc. 

If in an harmonic pencil one ray halve innerly the ^ between 
two conjugates, the other (its conjugate) must halve the same ^ 
outerly; i.e., unust be _L to the first. 



54 CO-ORDINATE GEOMETRY. 

Clearly, both these propositions may be converted by simply 
exchanging the words innerly and outeiiy. 

Also, if of two conjugates in an harmonic range one he at co^ 
the other halves innerly the tract betiveen the other pair; and, if 
one pair of conjugate rays in an harmonic be at H. A, they halve 
innerly and outerly the ^ of the other pair. 

EXERCISE. 

Prove by similar A that {12 3 4}= {!' 2' S' 4'}. 

42. Cross Ratio of four rays given by their equations. 

Be the base-ra3^s, N^ = 0, iVg = 0, a pair of conjugates, and 
be JV2 = ^i — ^2^3 = O5 iV4 = iVi — ^4^3 = 0, the. other pair. 
Denoting the rays b^^ their proper indices, we see at once, from 
Art. 32, 

. sin 12. sin 1 4 , c 1 o o ^ > x n 

A2 = , A4 = ; hence ) 1 2 3 4 j = Ag : A4. 

sin ^3 sin 43 

If Li = 0^ ^3 = be the base-rays^ /i, /g their normalizing 
factors; then Xi = 0, jLj — Agig^O, Z/3 = 0, ii — A4X3 = 
may be written in ipormal form ; thus, 

fLz—^i fiLi — X^^'fLs = 0', 

whence, by the above, we have again, J 1 2 3 4| = A2 : X4. 

If L'= 0, L"== be base-rays; then to find the CR. of any 
rays, i'-Xii" = 0, i'-Agi''^ 0, i.'-A3X"=0, X'-A4i"=0, 
take either pair of conjugates as base-rays, say first and third, 
and express the other pair through them ; thus, L'—XiL"=Li, 
L' — XqL" = Ls; whence, finding L' and L", and substituting, 
we find the other pair are 



CKOSS BATIOS. 



55 



A-^^^^^A^O, A- ^i~^^ 



A3 — A 



A3 — A4 



i3 = 0; 



hence, 



1 2 3 4| 



_ Xi — A2 • A3 — A4 



^3 ' ^4 — K 



43. If A2:A4 = — 1, the fjencil is harmonic; that is, spe- 
cially, L'=0, L"=0, and L'-XL"=0, L' + XL"=0 
are two harmonic pairs. 

The four rays (1, 2, 3, 4) of the pencil (i', L") are har- 
monic when, and only when, 



Aj — A2 • A3 — A4 

'^2 — ■^s * A4 — A^ 



-1; 



i.e. , when AiA3 — | . A^ -f A3 • A2 + A4 + Ag A4 = 0. 

44. If A, B and P, Q be two pairs of harmonic points, then 
for P midway between A and 5, Q is at 00 ; as P moves out 
from its mid-position toward 5, Q moves out from its mid- 
position at 00 into finity toward B. As P falls on jB, so does 




Q. The same remarks hold when A is put for B ; also, when 
rays SA^ etc., are put for points, it is necessarj^ only to note 
that for Q in 00 SQi^ \\ to the carrier of the range. 

If the CR. =+1, clearly a pair of conjugates (rays or 
points) must fall together. 



56 



CO-OKDINATE GEOMETRY. 



45. If A = 0, ^2=0 meet at /S', and X/^ 0, is'^ 
at S', any ray, L^ — XL2 = 0, of the one pencil is said to 
correspond to the ray, Li—kL.2 = 0^ of the other ; the pen- 
cils are called homographic. Since the CR. of two pairs of 
rays depends only on their parameters, the X's, it follows 
that : 

The CR. of two pairs of rays equals the CR. of the correspond- 
ing pairs in any other homographic pencil. 

By eliminating X we get LxLii — L^^x "= j this, then, is a 
relation between the Cds. (ic,2/) ^^ ^^J junction-point of two 
corresponding rays ; since the i's are of first degree in x and y^ 
this Eq. is of second degree in x and y ; hence, 

The junction-points of pairs of corresponding rays in two 
homographic systems ^pencils) lie on a curve of second degree. 



EXERCISES. 

1. iVj==0 and iV2 = enclose an Y a; find the Eqs. of the ±s to 
them through their intersection (iV^, N,^. 

2. How do N-^ — xK^^O and xNy — N.^ = {) He in the pencil {N^, N^)l: 

3. Show that the transversals from the vertices of a A to the contact- 
points of the inscribed resp. escribed circles meet in a point. 

4. iVj + Aj iV^2 = ^^<1 -^1 + ^^2 ^2 = being taken for base-rays in 

N-^+ \N^=zO, what is the 
value of h when N^+k'N.j.= 
is the same as N-^_ -\- A^iV^ 

5. Is 2,x- 10i/+4=0 
of the pencil 5 a: — 7?/ + 3 
4-A(2:r + 3y-l)=30? If so, 
express its Eq. through 
1x— 4?/+2 = 
and 19x + 14^ — 4 = 
as base-rays. 

6. The CE. of a pencil is r; three rays are L^=0, L^-=0, L^-^kL^=0; 
what is the fourth 1 E.g., llx-2?/ + 7 = 0, ^x-^by = Q, 17^ = 2a:-}- 25,. 
r=9:8. 




INVOLUTION. 57 

7. Four rays of the pencil 5a:— 7j/ + 3 + A(2x + 3_y — 1)=0 are 
llx-\-2t/^0, x+o^lS2j, lx + 2=4:y, 16x-j-Sy = 2; find the CR. 
{First find A^, Ag, A3, A^.} 

8. The inner halvers of the Ys of a A are cut harmonically, each by 
the centres of an escribed and the inscribed circle. 

9. Are these two pairs of rays harmonic : 

3y-\-4:X = 25, 2y-3x=ll; 7j/-2a: = 47, 10r-^ = 3? 

10. Find the fourth harmonic to 

11. Find the harmonic conjugates to each of the three rays in the 
pencil Z/+AiZ"=0, L'+\.,L"=0, L'-\-\^r'=0. E.g., L'=2x+3>/-^, 
L"=7x — 2tj-\-l; takeasrays, 13y-3Sx=10, 23x-3j/ = 2, 9x+3/ = 4. 



Involution. 

46. To any one pair of conjugates there is an infinity of har- 
monic pairs of conjugates ; for the Eq. that says the pair 
(Ai, Ki) is harmonic to the pair (A, k) : 



2 Ak - A + K . Ai + ki + 2 AjK-i = (1) 

is clearly fulfilled for an oc of values of A and k. If a second 

pair (A2, Kg) be also harmonic to tbe same pair (A, k), then 
must hold the second Eq., 



2AK-A4-K • A2+K2 + 2A2/C2 = 0. (2) 

These two Eqs. are linear in Ak and A + k; hence they are 
both satisfied by one, and only one, pair of real values of Ak 
and A + K ; this pair will 3'ield one, and only one, pair of values 
of A and k, which may be real or imaginar}^ ; hence. 

In any ijencil there is ahcays one, and only one, j9a^?' of rays, 
real or imaginary, harmonic to each of two given pairs. 

47. If, now, there be a third pair (A3, Kg) harmonic to the 
same pair (A, k), then must hold the third Eq., 



2Ak — A + K . A3+K3 + 2A3K3=:0. (3) 



= 0, 



58 CO-ORDINATE GEOMETJEIY. 

These three Eqs., (1), (2), (3), will be fulfilled by the same 
pair of values (A, k) when, and only when, 

1 Ai -j- K^ Xi Ki 
1 Ag + K2 As K2 
1 A3-I-K3 A3K3 

which is therefore the JEq. of condition declaring that the three 
pairs of rays, I.'-Aii"=0, L'~k,L"=0', L'-X^L'^^O, 
X'-K2i."=0; i'-A3i"=0, i'-/c3i"=0, have a com- 
mon harmonic pair. 

Three or more pairs of rays harmonic each to the same pair 
form an Involution. The common harmonic we may call focal 
rays. Any transversal is cut by an involution of rays in an 
involution of points^ whose foci are the section-points of the 
focal rays. 

Pairs of conjugate points correspond to pairs of conjugate 
rays. The foci and a pair of conjugate points form an har- 
monic range. The mid-point between the foci is called Centre 
of the Involution. 

48. The p)roduct of the central distances of a pair of conjugate 
points is a constant : the squared half-distance between the foci. 

F C F' 

P' ' P 

For \FPF'P'l = FP' F'P'-.PF'- F'F=-1; itFF'=2c, 
CP=d, CP'=d', then (c+d)-(d' -c) = - {c-{-d')-{d-c), or 
dd'=c'. 

If P fall on F or F\ so must P' ; i.e., foci are double points, 
and focal rays are double rays. If P and P' fall on the same 
side of (7, d and d' are like-signed, c^ is -f-, c is real, the foci 
and focal rays are real ; but if P and P' fall not on the same 
side of (7, d and d' are unlike-signed, c^ is — , c is imaginary, 
the foci and focal ra3"'S are imaginary. 

The foci fix the centre, and so the Involution ; but, by Art. 
46, two pairs of conjugates fix the common harmonic pair ; i.e., 
the focal rays resp. points ; hence, two pairs of conjugates (ra3's 
resp. points) fix an Involution. 



DIAGO:^rALS OF A FOUB-SIDE. 



59 



49. If L'=0, L"=0 be focal rays, L'-X,L"=0, where 
s=|l, 2, 3, 4|, am' four rays, then theu* conjugates are 
L' + X,L"=0 (Art. 43), and the CRs. of the two sets are 
plainly equal ; hence. 

In any Involution the CRs. of any four elements and of their 
conjugates are equal. 

Cor. If six points (or rays), A^ A', B, B', C, C", be in invo- 
lution, then 

\ABCA'l = \A'B'C'Al, 

which tests whether the thh'd pair be involved with the other two. 

Cross Ratio and Involution are of greatest import to Higher 
Geometry and Mechanics. Minuter treatment were out of place 
in this elementary work ; but the foregoing, it is hoped, may 
excite the reader's interest, and incite him to further pursuit of 
the subject. A single illustration is added in the proof of the 
familiar theorem : 

Each diagonal of a four-side is cut harmoniccdly by the other 
two. 







/ 




\i/ 




/ / 


\A. f ^/^ 




• / 
r / 


\ '1^^^^^ 




'-^ / 


I>^' 




^ ^^ 1 ^ 


/ \ 




• '^^y^ 


/ \ 




y ^7^ 






/ ^"' 


'^P \ 




'\^ / 


"^^^ \ 




y ^ / ' 






^ .y^ 1 f 


^ ^ \ 




' ^ h 


^^ \ 


A'^ 


C^ r 


"Ac' 




/ / 
/ / 


^ \^^.^ 



Be AB, AC, A'B, A'C the sides, AA', BB', CC, the 
diagonals. Then 

\ CDC'D'l = Al CDO'D'l = \ CI'B'A'l 

= D\CI'B'A'l = lC'IBA'\ 

= A\ C'IBA' \ = \ C'DCD'l 

= 1: ICDC'D'l ; .\\CDC'D'l = ±l. 



60 CO-ORDINATE GEOMETRY. 

But a CE,. can =1 oul}' when two rays fall together; hence, 
Q.E.D. Conduct the proof for the other diagonals. 



Equations of Higher Degree representing Several 

Right Lines. 

50. If Ly = 0, ^2= 0? •••? -^« = be Eqs. of n RLs., their 
product Li' L2 •'• Ln== will be of nth degree in x and?/, 
and will be satisfied by all and only such pairs of values of x 
and y as reduce some factor, as L^, to 0; i.e., the Eq. 
L1L2'" L,^ = will picture all and only such points as lie on 
the RLs. A = 0, X2 = 0, .•.,i„ = 0. 

If all the jL's be homogeneous in x and y, so will be their 
product, and not otherwise ; but then all the RLs. go through 
the origin ; .'.nRLs. through the origin are pictured by an Eq. 
of nth degree, and homogeneous in x and y. 

Conversely, such an Eq. pictures n RLs. through the origin. 

For, on division by cc*", the Eq. takes the form 



co + ci -^ +C2^+ ••• +c, ^4- ••• -he,, y^=0. 



This Eq. of 9ith degree in the ratio y : x has n roots, s^, s^, Sg, 
., s^, and may be written 

This Eq. is satisfied when, and only when, a factor equals : 

y 
as 8,^ = 0, or y = s^^x. But this is the Eq. of a RL. through 

the origin, and there are n such factors. 

The RLs. are real or imaginar}', separate or coincident, 
according as the roots, the s's, are real or imaginary, unequal 
or equal. 

51. If the Eq. be not homogeneous in x and ?/, we may test 
whether it be resoluble into factors of first degree in x and y 



PAIRS OF EIGHT LINES. 61 

by assuming such a resolution, forming the product of the 

assumed factors, and equating coefficients of like terms in x and 

y in the assumed and given expressions. For an Eq. of nth. 

^ , ^j, I 

degree there must then hold Eqs. of condition among 

J. • ^ 

its coefficients. Our immediate concern is only with the Eq. of 
second degree, which may be written thus : 

hx" + 21ixy +jy' -\-2gx-^ 2fy+ c = 0. (1) 

Instead of the tedious general method we may employ the 
following, especially as its incidental results are useful : 

Pass to II axes through the section-point {x^^yi) of the two 
RLs. which the above Eq. is supposed to represent. This is 
done b}' putting x -\- x^ for x, 2/ + 2/i ^ c>r y ; on collection there 
results 

kx" + 2hxy -^jy'^ + 2g'x + 2fy + c'= 0, (2) 

where g' = kx^ + hy^ + ^, /' = lix^ +jyj_ +/, 

c' = kx^^ + 2 hxiyj_ +jy^ -\-2gx^^ 2fy^ + c. 

This result is got by reasoning thus : terms not containing x^ 
or 2/i are found by supposing a7i = 0, 2/1 = 0, which gives the 
original expression ; terms not containing x or y, by supposing 
x = 0, y = 0, which gives the original expression with %, ?/i writ- 
ten for X, y ; terms containing a subscribed and an unsubscribed 
letter can result only from the original terms of second degree, 
appear doubly, and are sj^mmetric as to the subscribed and 
unsubscribed letters. This reasoning gives the following as the 
result of the substitution : 

kx^ + 2 hxy -\-jy' -\-2gx-\-2fy + kx^- + 2 hx^y^-^jyi^+ 2gx^ 
+ 2/i/i + 2kxx^ + 21i{x^y + y^x) + 2jyy^ + c = ; 

and this collected gives Eq. (2). It is important that the stu- 
dent thoroughly master this argument. 

Now, if Eq. (1) is a pair of RLs. through (0^1,2/1)5 Eq. (2) is 
a pair through the origin ; hence, (2) must be homogeneous of 
second degree in x and y ; hence, terms of lower degree must 



62 CO-OEDINATE GEOMETKY. 

vanish ; hence, p''=0, /'=0, c'=0. But c' may be written 

and the coefficients of x^ and 2/1, being g^ and/', are ; hence, 
so is gx^+fyi + c, hence, between x^ and 2/1 must consist the 
three Eqs., 

kx^ + liyi + P' = 0, /iiTi + J2/1 +/= 0, ^Xi +/?/i + c = 0. 

The condition of this consistence is 

= 0. 



k 


li g 


h 


Jf 


g 


fc 



This Determinant is named Discriminant of the Eq. of second 
degree, and is denoted by A. Hence, A = is the condition 
that the Eq. of second degree represent two RLs. 

The co-factor of any element of A will be denoted by the cor- 
responding capital letter. The Cds. (i»i,2/i) of the section of 
the RLs. ma}^ be found from any two of the above three Eqs., 

and are Xi = G:C, yi = F:C. If (7, or 7cj — h^, be ^ 0, 

Xi and yi are finite, the RLs. meet in finity ; if (7=0, Xi and 
2/1 are 00, the RLs. meet in 00 ; i.e., the RLs. are j]. 

52. It is to note that changing the origin but not the axial 
directions does not change the terms of highest, i.e., of second, 
degree. 

To find the direction coefficients, it suffices to factor 

Jcx^ + 2 hxy -\-Jy^, 

or y^-^2-~ xy -{- -y^. 

J J 

If the factors be y — s^x, y — Sgcc, 
then 



wnence, 



Sl+S2 = 


-2\ 
J 


SIS2 


J 


Si — S2 = 


^■Vh^- 
J 


-kj 





EQUATION OF BISECTORS. 63 



?^ = J-7i+V/i--A:jJ :J, 



Hence, by Art. 26, if the RLs. enclose an ^ ^, 



tan </) = 2 -yhF—l^j • sin w : ['k-\-j — 2h cos w J , 



or tan c^ = 2V/i^ — kj : {^^ +ij ; 

the RLs. are J_ when 

k -j-j — 2h cos CO = 0, or when k + J = 0, if w = 90°, 
and are imaginary when h^ — kj<.0. 

53. To find ^/lepaiV of RLs. halving the ^s bettveen the pair 
kx^ -\- 2 /icc?/ -i-jy^ = 0, or 2/ — SiO? = and y — S2X = 0. 
Brought to the N.F., these Eqs. are 



y — SiX: Vl + si^ = 0, y — S2X : Vl + s,^ = 0. 

Their sum resp. difference is the Eq. of the outer resp. inner 
bisector ; and the product of these is 



y — SiX : 1 -\- St_^ — y — S2X : 1 + §2 = 



or Si — 82^ ' y^ — 2 ' S1S2 — 1 ' Si — S2' xy — s^ —S2'X^ = 0., 



Divide by s^^ — Sg, replace s^ + So, s^s^ out of Art. 52 ; whence, 

2/^ -1 ^ x?/ — ic^ = 0, the Eq. sought. 

h 

Note that this pair of halvers are always real, though the pair 
kx^ -\- 2hxy -\-jy^=z 0, whose ^s they halve, maj^ be imaginary. 



54. The general Eq. of second degree is not hcmogeneous in 
X and y ; we may make it so by multiplying the terms of lower 
degree by fit powers of some linear function of x and y. So 
we get 



64 CO-ORDIKATE GEOMETRY. 

kx" + ^lixy +jif + 2gx + 2/?/ + c = 0, (A) 

lx-\- my = 1, (B) 

kx^ + 2 Jixy -{-jy'- J^(2gx-{- 2fy) (Ix + my) 

+ c{lx-}-myy = 0. (C) 

By Art. 50, (C) represents two IlLs. through the origin, and 
by Art. 30, they go through the intersections of (A) and (B). 

Note the method of this article, as it solves the problem of 
finding the Eq. of the RLs. through the origin and the section- 
points of a E.L. with a curve of any degree. 

EXERCISES. 

1. What RLs. are pictured by the Eqs. : 

(a) X — c .X — d = 0', (f) 4_y2 — 15x_y — 4a;2 = 0; 

{h)xy = 0; (g) 7/2+13x^-10:r2 = 0; 

(c) x2-4i/2=0; (h) ?/2_2t^ + 3^2 = 0; 

(d) T^ — ?/2 = ; (i) y^ — 2xi/ sec -{- x"^- = ; 

(e) x2-5r//4-4?/2=:0; (j) x'- + xi/ -6i/'' + 7 x + 31 y -1S = 'i 

2. Eind the Ys of the above pairs, and their bisectors. 

3. Eor what values of \ do these Eqs. picture R-Ls. : 

(a) 12a:2-105:j/ + 2?/2 + ll^c-S^/ + A ^0 ; 

(b) 12x^-\-\xy +2?/2 + ll-5?/ +2 = 0; 

(c) 12x^-\-S6xy -{-Af + 6x + 6?j + S=0l 

4. Show that the ELs. joining the origin to the section-points of 
Sx'^ + 6xi/ — 3f + 2x + Sy = and 3:r = 2y+l are ±. 

5. N^ =r 0, iV2 = 0, N^ = being sides of a A, find the RLs. : 

Rectilinear Loci. 

55. The sum total of positio)is to ivJ/ich a point is astricted by 
some geometric coyidition is called the locus of the point. 



THE EIGHT LINE AS LOCUS. Q5 

To express the condition through constants and the current 
Cds. of the point is to find the Eq. of the locus. 

For doing this no fixed rule can be given ; each problem or 
class of problems must be solved for itself. 

In general, the expressions will be made more simple by 
choosing the axes with special reference to the figm-e of the 
problem, but more symmetiic by, avoiding such reference ; 
sometimes the one advantage, sometimes the other, will out- 
weigh. Often the point is fixed as the junction of pairs of 
corresponding lines in two systems of lines ; the common par- 
ameter of the two systems must then be eliminated. Thus, if 
the point lie on the curve F(x^ y • p^ = 0, and also on the 
curve ^(cc, y;p) = 0, by giving j9 any value we may find a 
pair of values of x and y satisfying both Eqs. ; i.e., we may 
find one position of the point ; but by eliminating j) we find a 
relation between ei'eri/ pair {x,y) which satisfies the two Eqs., 
whatever p may be ; i.e., we find a relation between the Cds. 
of the point in any position ; i.e., we find the Eq. of its locus. 
There may be more than one parameter ; the number of 
Eqs. needed is, in general, one greater than the number of 
parameters. 

EXERCISES. 

1. A point moves so that the sum of its distances from the sides of an 
^ is constant ; what is the point's path ? 

If /jX 4- m-^ij + n^ = 0, l^x + m^t/ + ^2 = be the sides of the Y, tlien 
— -^ + - — . - — s IS the locus, a RL. 

The sides being axes, at once x + y =z s : sin w. Draw the RL. 

2. The sum of a point's distances from n RLs. is s; find its path. 

3. The ratio of a point's distances from two RLs. is fj-j^: fj.2; find its path. 

4. From side to side of a given RL. are drawn II tracts ; a point cuts 
them all in a fixed ratio ; find its path. 

5. The ends of the hypotenuse of a right A move on the rectang. Cd. 
axes ; where does the vertex move 1 



Q6 



CO-OKDINATE GEOMETRY. 



6. Given one E. X fixed, 
another turning about a fixed 
point F ; find the locus of P 
cutting the junction-line of 
the sections of fixed and 
moving sides, in the ratio 

Hint. Take the fixed 
sides as axes, express OA 
resp. OB through x resp. y, 
and project on OF. 

7. Tracts are drawn from 
the origin to any point of a 
HL., y = sx ^ b ; find the 
locus of the vertices of equi- 
lateral As constructed on the 
tracts. 

Take the slope a of a 
tract to the JT-axis as par- 
ameter, and proceed thus : 

r/'— b : (1 — s cot a), 
t = y': sin a 
= b : (sin a — s cos a) ; 
x=t . cos (60° — a) = b cos (60° — a) : (sin a — s cos a) 
= 6 (1 4- VS . tan a) : 2 (tan a — s) ; 
y : a: = — tan (60° — a) = (tana— VS) : (1 + VS .tana) ; 
.•.tana= (xVS -{■ y) : {x — yVS) ; 
whence {y + sVS) + x (VS^— s) —2b, 
a E.L. sloped 60° to the given EL. Draw it. 

8. Find the locus of the 
intersection of ±s to the 
sides of a A, cutting the sides 
at points equidistant from 
the ends of the base. 

Take the base as X-axis, 
either end as origin ; the 
Eqs. of the sides are y — s^x, 
y —s^x -{■ b; take the dis- 
tance d as parameter ; the 





EXERCISES. 



67 



Eqs. of the ±s are y — ar + Cj, y— x ■\- c^) express Cj, c^ through 



d, so we find the ±s are s^y -\- x = dVT+sJ, s<^y -\- x -\- 
Hence eliminate d. 



-d\/l-\-s^'. 



9. Eind the locus of the intersection of ±s to the sides of a A through 
the points where they caut II to the base. 

10. On the sides of a given Y are laid off from its vertex tracts whose 
sum is s ; ±s to the sides through the ends of the tracts meet at P. 
Where is P ? 

11. Given a vertex, the directions of the sides, and the sum of the 
sides, of a parallelogram ; find the locus of the opposite vertex. 

12. Find the locus of the intersection of RLs. joining crosswise the 
points where pairs of rays through 
a fixed point cut the sides of a 
fixed ^. 

If the axial intercepts of the 
rays be a^, \, a,^, \, the cross lines 
are 

X V ^ X V ^ 

- + f = l, - + ^=1; 
if (zj, y-^) be the fixed point, 

Form the difference of the first, and also of the second, pair of Eqs. ; their 

quotient gives f = _ •^, the locus sought, which is seen to be the same 

H 
for all positions of P^ on the RL. OP^. 

In the following problems, about a A ABC^ take AB as the 
+ X-axis of rectang. Ccls. 

Five noteworthy points of a A are : mass-centre (intersection 
of medials), orthocentre (intersection of altitudes), centre of 
vertices (or of circumscribed circle), centre of sides (or of 
inscribed circle), intersection of transversals from vertices to 
contact-points of opposite escribed circles. 

13. Find the loci of these points for similar A with a fixed Y A. If r 
be the radius of the circumscribed circle, the sides of the A are 2r . sin^, 
2r . sin P, 2r . sin C ; the Cds. of the points (in order) are : 




68 



CO-OEDINATE GEOMETRY. 



T = t] r (sin A cos B 
X —2r cos A sin B, 



2 cos A sin 5), 



X — r sin A + B, 
A 



X —4:r cos 



sm 



B 



cos 



^ + 5 



y = J r sin A . sin 5 ; 
y — 2r cos J. cos B ; 

y — —r cos ^ + J^; 

. . J. . i? 

y =^ 4 ?• sm — • sm — 

^ 2 2 



cos 



2 ' 



y 



I ?• sm 



A' 



sm 



i>'" 



r = 2 r (sin ^ — sin 5 + cos A . sin B), 

Eliminate the parameter r from each pair ; it is seen that as the A 
swells, the five points move out on five ELs. through the centre of 
similitude. 

14. rind locus of centre of sides, only A and its counter-side a constant. 

15. Find locus of orthocentre, only B and h constant. 

16. Eind loci of first, second, third, fifth points, only B and a constant. 

17. Find loci of the first four points, only A and h constant. 

18. Find loci of the first four points, only A and c constant. 

19. Find loci of third and fourth points, only C and h constant. 

20. Find locus of mass-centre when c is constant in size and position, 
while C moves on the RL. ^= sx + n. 

21. Given an Y of a A and the sum of the including sides ; find locus 
of P cutting third side in a fixed ratio. 

In problems about tracts of changing length and direction, 

measured from a fixed point, polar Cds. are recommended. 

p 

22. Chords through a fixed 

point of a circle are produced 
till the rectangle of chord and 
chord produced is constant ; 
find locus of end of produced 
chord. Take the diameter 
through the fixed point as 
polar axis ; then OP — p, OC 
= dcose, OCOP^k"^; pcose 
= Jc^:d, a RL., as is also clear 
from similar A. 

23. Two tracts whose 
lengths are in a fixed ratio 
enclose a fixed Y at a fixed 
point, and the end of one 
moves on a EL. ; how moves 
the end of the other 1 



J^^ 




^, \! 




C?^ ■■' 








y 


/ 


/" 


A 


A^ 


i<= 


-^^^^^ 


D 


e?" — '■ 










FAMILIES OF BIGHT LINES THKOUGH A POINT. 69 

Take as polar axis the J_ from the fixed point on the fixed RL. ; be OA 
and OP the tracts, and OP:OA—r. Then p — rp sec (9 ~ a), a RL. 
Draw it. 

24. Erom a fixed point is drawn a ray cutting two fixed RLs. at A 
and A', and on it P is taken 

so that OP is the harmonic Y/) 

mean of OA and OA' ; find 
locus of P. 

OX being taken as polar 
axis, the Eqs. of the fixed 
RLs. DA and DA' are 

1 : p = / cos 6 -{- m sin 6, 
1 : p = I' cos + m' sin ; 

these Eqs. hold for the same 

6 where p is OA resp, OA' in V 

the first resp. second. By hypothesis, 2 : OP — 1 : OA + 1 : OA'. Writing 
p for OP, we get 2: p— {l-\- l')cos d -{- {m + m') sinO, the Eq. of a RL. 
through D. 

25. Generalize Ex. 24 by taking n instead of 2 RLs. through D. 

26. Given base and difference of sides of a A ; at either' end of base is 
drawn a JL to the conterminous side ; find locus of its intersection with the 
inner halver of the vertical angle. 

Families of Right Lines through a Point. 

56. Thus far in each problem have been two conditions, 
enabling ns to determine the two parameters in the general Eq. 
of a RL. Had there been but one condition in the problem, 
the result would have contained one parameter undetermined, 
and so would have represented a family of RLs. Should a 
parameter appear in a result linearh^, then, b}^ Art. 29, all RLs. 
of the family would pass through a point. When the parame- 
ters appear linearly, both in the general Eq. of a RL. and in 
some Eq. of condition, all RLs. of the famil}' go through a 
point ; for, by help of the Eq. of condition, we may eliminate 
one parameter and leave the other appearing linearly. 

Such is the case, e.g., when the parameters are the current 
Cds. x\ y' of a RL. ; for then they fulfil some linear condition, 
lx'-\-my'-{- n = 0. 



70 CO-ORDINATE GEOMETRY. 



EXERCISES. 

1. The vertical Y and the sum of the reciprocals of its sides are given 
in a A ; find the Eq. of the base. 

Take the sides as axes ; then, the reciprocals of intercepts of the base 
on the axes being I and m, the Eq. of the base is Ix + my = 1 ; also, 
^ -f m = c. Hence the base turns about a fixed point. Find the point. 

2. If, as the vertices of a A move on three fixed RLs. through a point, 
two sides turn about fixed points, the third turns about a fixed point. 




Take two of the fixed RLs., as OA, OB, for axes ; then y = sx is the 
third RL. OG. Be {x^,y{), (^2' ^2)' the fixed points, {x', y') the third vertex 
C; then y'=sx', and the Eqs. of AC and BG are 

(^1 — ^')y- iVi - soo') x+x' [y^ - sx^) = 0, 

(^2 - ^') 3/ - ih - ^^') ^ + ^' (^2 - 5^2) = '^• 
Hence, find OA, OB, and form the Eq. of the third side AB, 

x{y^-sx') : (y^ — sx^j-y (xj^-x') :{y^- sx-^) =x'. 
The parameter x' enters this Eq. linearly ; hence AB turns about a 
a point. Find the point as the intersection of two base RLs. by solving 
the Eq. as to x' . 

3. All RLs., the sum of proper multiples of the distances of n fixed 
points from any one of which equals 0, form a family through a point (the 
centre of proportional distances of the points) . 

Be {xjc, yjc) one of the points, /Xk the proper multiplier, x cos a + ?/ sin a 
—p::=0 one of the RLs. Then, by hypothesis, 

Jc=n lF=n k=n 

-^ fikXjc COS a + ^ juj^yjc sin a — p -^ iiijc = 0. 

*=1 k=l k=l 

Between this Eq. and the Eq. of the RL. eliminate p ; on division by 
cos a (which, and sin a, may be written before the summation sign 2) tan a 
appears, as parameter in the result, linearly. By proceeding as directed in 
Ex. 2, the fixed point is found to be 

(Sa^a^a: 2;Us, ^fxuyu'-'Sfijc)' 



THE CIRCLE. 



71 



CHAPTER III. 



THE CIRCLE. 

Before treating the general Eq. of second degree, it may be 
well to treat a special case of great importance. 

57. By Art. 15, if r be the distance between (a?, y) and 
{x^, 2/i) 5 then 

X — x{ + y — y{ + 2 • X — Xj^ ' y — y^' cos w = r^. (I') 

Hold r and (a^i, 2/1) fast, letting (x, y) vary ; (:c, y) will keep 
r distant from (x^, y^) ; the sum total 
of its positions is called a Circle, with 

radius r, centre (Xi, y^) . Hence (I') / yp^'y 

is the rectilinear Eq. of a circle. 

For o) = 90° the important rectang. 
Eq. is 

° (I) 



X 



^1 +y — yi = ^"• 



For iCi=0, ?/j = 0, i.e., for the 
central Eq., the simpler forms are 




and 



a? -\- y^ -{- 2 xy cos co = ?* 



(IF) 



58. A circle is known completely when are known its radius 
(in length) and its centre (in position). Since in (I') resp. 
(I) radius and centre are expressed generally', it is clear that 
(I') resp. (I) is a general form to which any Eq. in oblique 
resp. rectang. Cds. of any circle may be brought. We note 
that the coefficients of x'^ and y"^ are equal : each = 1 , and the 
coefficient of xy is 2cosa) resp. 0. Divided by h the general 
Eq. of second degree takes the form 



72 CO-ORDINATE GEOMETRY. 

aj2 4_ 2 - iC2/ + -2/" + 2 ^ -f 2-^ 4- - = 0. (Ill') 

k k k k k 

Accordingly, if this be the Eq. of a circle^ then must j :k = l 
or j = k, and /i : A; = cos w or h = k coso). 

These, then, are conditions necessary that the Eq. of second 
degree picture a circle. They are also sufficient^ for where they 
are fulfilled, the three arbitraries x-^^ ?/i, r, may be so chosen as 
to satisfy any set of values of the three coefficients, 

t k ^ k^ k 

Two problems may now be solved : 

1. Given centre and radius, to form the Eq. of the circle. 
Substitute in (I') resp. (I), and reduce. 

2. Given the Eq. of the circle, to find centre and radius. 

Divide the Eq. by the coefficient of x^, equate the coefficients 
of X and y, and the absolute term, to their correspondents in (I') 
resp. (I), and so find a?i, yi, r. 

EXERCISES. 

1. The centre of a circle is (3, —4), radius 6 ; find its Eq. 

2. Find the Eqs. of the circles whose centres and radii are (0,0), 9; 
(7,0),3; (0,-2), 11; (-4,17), 1. 

3. Can these Eqs. picture circles, and of what radii and centres : 

15:^2 + 153/2+ 15x^-90 ?/- 45 r=0; 
3^2 - 3x^ + 3/ - 9a: + 12 ?/ - 10 = ? 

4. Find centres and radii of Ta;^ — 7?/2 + 49 a: + 84_y + 14 = and 
5^r2 + 5/ + 25a: - 15 ?/ + 20 = 0. 

59. As is clear on comparing (I') and (HI'), the general 
p]qs. for determining Xi, y^, r, are 



DETERMINATION OF THE CmCLE. 73 

^1 + 2^1 cos w = — g : Jc, 
2/1 + % cos CO = — /: /j, 
a?!^ + 2/1^ + 2 Xi^i cos (jo — r^ = ciJc; 

or, for rectang. axes, more simply, 

x^ = — g:Jc, 

2/1 = -/ : ^5 

a?!^ + 2/1^ — ^"^ = c : ^ ; 

The equivalent forms, in which the coefficient of ic^ + 2/^ is 1, 



X — x{^ + y — y^^ — r^ = 0, 
or x^-j-y'- + 2gx-{-2fy-}-c = 0, (III) 

may be called the Normal Form (N.F.) of the rectang. Eq. 

In the N.F. Xj^ = -g, yi = -f, 'i^ = g^ -^f^ -c; i.e., the 
Cds. of the centre of a circle are the negative half-coefficients of x 
and y in the N.F. of its rectang. Eq. ; the squared radius is the 
sum of their squares less the absolute term. 

The circle is a real, a point-, or an imaginary, circle, accord- 
ing as 

^^+/^-^c>0, or =0, or <0. 

The Cds. of the centre do not contain the absolute term; 
hence, if this alone change, the centre does not change ; i.e., 
circles whose JEqs. differ only in absolute terms are concentric. 

If the absolute term be 0, the curve goes through the origin. 

60. The Eq. of a circle contains three arbitraries, x^ y^ r, or 
g, f, c. Hence, three conditions are needed and enough to fix 
a circle. To find a circle fixed by three conditions, express 
these through Eqs. and thence find the arbitraries. Thus, 
find the circle through the points (1,2), (3,-5), ( — 2,1). 
Since each Cd. pair satisfies the general Eq. 

x' + y'-^2gx + 2fy + c = 0, 



74 



CO-aRDINATE GEOMETEY. 



we get 2^ + 4/+c + 5 = 0, 

6^-10/+c + 34 = 0, 

whence, finding {/, /, c, and substituting, we get 

2S{x^ + y')-2dx + 87y- 520 = 0. 

Or, by Determinants, more neatly, thus : If the circle 
x' + y'+2gx-^2fy-\-c = 

goes through (x^, y{) , (a^g, 2/2) , (^&, 2/3) , then 

o^,'-\-y.' + 2gx,-}-2fy, + c = 

for A:= 1, 2, 3. These four Eqs. consist for the same values 
of ^, /, c when, and only when, • 

x^ -\-y" X y 1 
^1^ + 2/1- ^1 Vi 1 

•^2 ~r 2/2 ^2 2/2 1 

. 2 I 2 1 

^3" ~r 2/3 ^3 2/3 -•- 

which is therefore the Eq. sought. Clearly, it is also the con- 
dition that four points, P, Pj, Pg, P3, lie on a circle. 



EXERCISES. 

1. Find the circles through (0,0), (a,0), (0, b) ; (a,0), (-a, 0), (0, b). 

2. If (^1, ?/i) (^2'y2) he ends of a diameter, the Eq. of the circle is 



or, for oblique axes. 



X — x^' X — X2 + y — yj^' y — 1/2 + [x — x^- ij—y^ + x — x^- y — tji}eoscc=0. 

N.B. The following familiar propositions in the Theor}^ of 
the Quadratic Equation are here recalled once for all : 

Be C2X^ -\- 2 CiX -{- Co = the general Eq. of second degree 
in X, Ti and ?2 its roots ; then 



DETERMINATION OF THE CIRCLE. 75 

(1) ri + r2 = -20i:(72, T,^T,= C,:C,. 

(2) For (7o = 0, oneroot = 0; for Cq = aDd (7i = 0, both 
roots = 0. 

(3) For 62=0, one root = 00; for C2 = and Ci=0, both 
roots = GO . 

(4) For (7i = 0, ?\ = — 7'2; i-e., the roots are equal, but 
unlike-signed. 

(5) For Ci = Co C2, the roots are equal and like-signed. 

61. To find where the axes cut a curve of second degree, 
equate y resp. a? to in the general Eq. ; so we get 

Tix^ -\- 2 gx -\- c = resp. jy'^ + 2/?/ 4- c = 0. 

The roots of these Eqs. are the intercepts on the X- resp. 
y-axis. (They are equal, i.e., the X- resp. y-axis meets the 
curve in two consecutive points, i.e., is tangent to the curve 
when, and only when, g^ — lie resp. f^='kc. See Art. 64.) 

Conversely, given the intercepts on the axes, to find the Eq. 
of the circle. Suppose Zc=J=:l, and be %, ag resp. &i, h^ 
the intercepts on the X- resp. Y-axis ; then 

2/--(&i + &2), 

c = a-^a^ — hxb2' 
Let the student interpret these Eqs. geometrically. 
Hence , x'^ -\- y^ — {ai-\- cio) x + 2fy -f- cf ^ as = , 

x'-^y' + 2^x - (61 + h)y -hhh= 0, 

resp. x^ -{-y^ — (% + 012) x— [b^ + 62)2/ + i-(c<!'i«2+ ^1^2) = 

are equations of a circle in terms of its intercepts : on the 
X-axis, on the Y-axis, resp. on both axes. 

N.B. Of course, only three intercepts can be assumed at 
will ; then the fourth follows from ajas = 6i&2' 



76 COOEDINATE GEOMETRY. 

EXERCISES. 

1. Where do the axes cut x'^+ij--\-bx— t/ — G — 01 

2. A circle touches each axis a from the origin ; find it. 

3. Find the equation of a circle through (0,2), (0,-4), (5,0). 

4. Find the equation of a circle referred to a tangent, and a chord 
through the point of touch. 

AYe have aj^ = b^ = b.^=0, a.^^ ±2r sin oj ; 

hence, x- -]- y"^ -]- 2 xi/ cos ca ±2rx sin co — 0. 

If the chord be a diameter, ca = 90° ; hence the important form 

x"^ -{- y^ ±2rx =: 0. 

Verify geometrically, and explain the double sign. 

Polar Equation of the Circle. 

62. Be d the tract from the pole to the centre, a its inclina- 
tion to the polar axis OD, r the radius. 




D 



Then the equation sought is 

p2 _ 2 f?p cos (^ - a) + ^2 = r2. 

The product of the two roots, pi = OPi, p2 = OP2', is con- 
stant, and = dr' — 7-2, a familiar theorem. 

The Circle in Relation to the Right Line. 

63. By Art. 16, a line of first degree (a EL.) cuts a line of 
second degree (as a circle) in two, and only two, points. 



CIKCLE AND EIGHT LINE. 77 

For, in the equation of second degree, put for y (say) its 
value sx-\-h (say) taken from the equation of first degree ; so 
we get an equation of second degree in x ; its two roots are the 
cc's of the two points common to the two lines. 

If u = {mn — ghn -i-fl^) : (l^ + m^) , 

and ^ = V§ {mn-glm-\-fiy-\- {2gln-Pc-n^) (P+m^) \ 

and {xi , ?/i ) be the common points of the RL. and the circle 

Ix-\-my-{-n = 0, a? + y' -{-2gx-\-2fy -\- c^O, 
the student may find (lower index going with lower sign) 

a^i,= -| + .y {u±v), y^^ = -{u±v), 

but he will not find any pleasure withal. To shun such labo- 
rious reckonings and such unmanageable formulas, we have 
recourse to special forms of the equations. 

Thus, the RL. y — sx^h meets the circle x^ -\- y^ = r^ m 

_^^__ ^ I (A) 

TheRL. a? cosa + ?/ sina =j9 meets the circle x^ -\- y^ = i-^ m 



fljj = p cos a ± sin a V'/'^ — IT 

2/j =z'p sin a q: cosaVH — j9^ J 

These pairs are real and different^ or equals or imaginary ; 
i.e., the common points are real and separate^ or consecutive^ or 
imaginary^ according as 

in (A) , 7-2(1 _|_ ^2) _ 52 is > 0, or = 0, or < ; 
in (B) , 7-2 -p^ is > 0, or = 0, or < 0. 

64. Real and separate points present no difficulty ; imaginary 
points have no existence in our plane, and are to be treated 



78 



CO-ORDINATE GEOMETRY. 




further on ; but consecutive points it is essential to his progress 
that the student understand clearly now and here. 

Coincident points fall together, have exactly the same posi- 
tion^ and are distinct only in tJiought, Thus, we may think of 

the intersection of two lines i, L' as 
made up of two points fallen together, 
and we may call it P or P' according as we 
think it belonging to L or to L'. Consec- 
utive points become coincident and in a 
particular way : by nearing each other on 
the same definite path (curve). Thus, if C be an}^ curve, P and 
Pi any two points on it, we may think P as fixed and Pi as 
taken at will nearer and nearer P; or, what comes to the same, 
we may think Pi as moving nearer and nearer to P along C. 
Be P a RL. through P and Pi. As Pi nears P, L turns about 
P, and the position of P is fixed completely by Pj. As Pi falls 
on P, L turns into some position T. Now P and Pi, thought 

simply as coincident^ cannot fix a 
RL. ; for they form but one point, 
and through this one point a RL. 
may be drawn in any direction. 
But P and Pi, thought as consecu- 
tive points (of C sa}'), do fix the 
position of L ; for, as consecutive 
points of (7, they become coincident 
by nearing each other along C. 
At every stage of this becoming- 
coincident^ L turns into a definite position, as Pi nears P; and 
at the end of it, the 5ez7i^-coincident, as Pi falls on P, it is left 
turned into a definite position T. It is specially to note that 
the being -GomoidQut of P and Pi has no power to fix the posi- 
tion of P ; it is only the particular way of their becoming coinci- 
dent that fixes it. As P/ becomes coincident with P (or P') 
in some other way : by nearing it along some other curve C", 
the RL. P' turns into some other position P'. 

Looking; at the algiebraic side of the matter, we find the 




CONSECUTIVE POINTS. 79 

likeness perfect. The two pairs of common roots {xi,yi), 
(cco, 2/2) of the two Eqs. y = sx-i-b, x^ -^ y^ = r^, are equal 
if 7-^= 5^ : (1 + s^) ; but not simply are they equal; they 
become equal in a particular way : not by b and s passing at 
random through any one of an infinite number of series of pairs 
of values up to that pair which makes r"^ = W : (1-f s^) , but 
by their passing through that particular series each one of 
whose pairs satisfies the two conditions, y^z= sx2-\-b and 
X.2 + yi = r^. The whole series of pairs of values of s and b 
being fixed, the last pair, which makes r^ = b^: (l + s-), is 
fixed; i.e., the last position of the E.L. y = sx-{-b through 
the consecutive points (x^, y^) , (ccg? 2/2) of x^ -\-y^ = r^, is fixed. 
Again, we see it is not the fact of being coincident, but the way 
of becoming coincident, which is significant. We further see 
that the concepts, coincident points and consecutive points, are 
not in themselves complete ; we think, though we do not always 
say : two coincident points of two curves ; two consecutive 
points of one curve. 

65. In the light of the above, we may now define a tangent 
to a curve as a RL. through tico consecutive points of the curve. 
Where the points fall together is called point of touch, contact, 
tangency. If for "RL." we put "curve," the definition still 
holds. We also see that the algebraic condition that a E,L. and 
a curve (or two curves) be tangent is, that two pairs of common 
roots of their Eqs. be equal. 

Hence, if y = sx-\-b touch x^-\-y^ = r^, 6^ = r^(l-f-s^) ; 
or. 

The ML. y = soc ±b\^l-^ s^ touches the circle ;x? -\-'kf — 7^ 
for all real values of s. 

This so-called magic equation of the tangent determines it by 
its direction (s) , not by its point of touch, and is useful in 
problems not involving this point. 

So, too, if x cos a-\-y sin a=p touch x^ -\-y^ = r^, 7^=1 p^. 
Hence, the Cds. of the point of touch are 

x-^ = r cos a, 2/1 = r sin a. 



80 CO-ORDINATE GEOMETRY. 

Substituting for cos a, sin a, p in x cos a-\- y sin a =p, we 
get 

The RL. ooooi + 2/2/1 = ^^ touches the circle ac^ -\-y^ = r^ at 
the point {xi,yi) of the circle. 

This equation determines the tangent by its contact-point 
(^15 2/1)1 and is useful in problems involving that point. 

66. The doctrine of Chords is so of a piece for all curves of 
second degree, that it is deemed best to state it here at once in 
full generality. 

We know how to find the intersections of a given RL. with 
a given curve ; the converse would be to find the RL. through 
given intersections ivith the curve. The general method is this : 

Combine the general Eq. of a RL. through two points with 
two Eqs. which say the given points lie on the given curve ; the 
result will be the Eq. sought. 

This tedious general method we may replace by special 
methods in special cases. Such is the following method of 
Burnside for the curve of second degree : 

Form an Eq. whose terms of degree higher than the first shall 
cancel (hence, it will picture a RL.), and which shall he satisfied 
by the Cds. of the points (x^^yi), (^252/2)1 only ivhen these points 
lie on the curve. 

Such an Eq. fovmed after this prescription is 

kx^ + 2 hxy -i-jy^ -{-2gx-{- 2fy + c 

= k(x-x^) (x-xo) -{-2h{x-x^) (y-y.) +j(y-yi) {y-y^)- 

This, therefore, is the Eq. of a secant through (a^i, 2/1)1 (^252/2) 
of the curve of second degree. The condition that this secant 
become a tangent is, that (^1,2/1)5 (^21272) f^H together. 

.'.kQ(? + 2hxy-[-jy'^^2gx-\-2fy^c 

= k{x - x^Y -\- 2 h{x - x^) {y - y{) -\-j{y - yi)\ 



EQUATION OF THE TANGENT. 



81 



or, after expanding, cancelling, transposing, remembering that 

kx^- + 2 hxr 2/1 + JVi + 2 ^a;i + 2/?/i + c = 0, 
finally, 

'kx^x^h{xT^y+y^x) -\-jyiy+g{x^+x) 4-/(2/1+2/) +c = (D') 
is the Eq. of a RL. tangent at (0^1,2/1) to 

lxXX + h{xy-\-yx)-\-jyy-{-g{x-^x) -\.f{yJ^y) +c = 0. (D) 

The Eq. of second degree being written thus, the Cds. appear 
in pairs, and we get the Eq. of the tangent by substituting for 
the first current Ccl. in each pair the corresponding Cd. of the 
Ijoint of touch. 

These general Eqs. of secant and tangent include all special 
cases, and are here deduced once for all. 

EXERCISE. 

Form by these methods, then simpHfy, the Eqs. of chords and tangents 
of the curves 

xy — c^, y"^ = 4:px, x^ •\- y^ — r^ ', [x — aY -\- [y — b)"^ = r^. 

67. From any point (x', y') may he draivn tivo, and only two, 
tangents to a curve of second degree. 

For, write F{x, y;x,y) = for Eq. (D) of Art. QQ ; then 
the Eq. (D') of the tangent is Fix^, 2/1 5 ^'5 2/) = ^ ; and clearly 
F{x^, 2/1 ; ^^ y) = F{x, y ; x^, 2/1) • If the tangent at (%, 2/1) go 
through (a;', 2/') , then i^(i«i, 2/1 5 ^\ y') = ^ '1 ^^icl? since (x^, 2/1) 
is on the curve, F(Xi,yi ; x^, 2/1) = 0. In the first of these Eqs. 
^11 yii ^' 1 y' all appear linearly; solving as to 2/1 (say), we get 
2/1 = an expression linear in a^^, x'. Substituting this in 
-^(%5 2/1 5 ^1' 2/1) == 0, which is of second degree in Xi and 2/1, we 
get an expression of second degree in x^. Solving this, we get 
two values of .x^, to each of which, since F{x-^,y^; x',y') = 
is linear in x^, 2/1^ corresponds one, and only one, value of 2/1 ; 
hence, there are two, and onh^ two, pairs of values (%, 2/1) ; i^e^j 
two, and only two, points of tangency. 



82 



CO-OEDINATE GEOMETKY. 



Of course these pairs may be real and unequal, or equal, or 
imaginar}' ; accordingi}^ the tangents will be real and separate, 
or coincident, or imaginary. The second case, of coincidence, 
arises when the point (x',y') is on the curve ; for, since the Eq. 
of a tangent through a point of the curve has been found uni- 
versally (see Art. 66), there is but one such tangent, which 
may, however, be thought as two fallen together. To tell when 
they are real, when imaginary, we may reason thus : 

68. Any curve, F(x,y) = 0, bounds off all points of the 
plane for ivhicJi F(x,y)<.0 from all points for ivhicJi 
F(x,y)>0. 



Y 



^(*^j/I^ 




<}- 



X 



For, assign y any value, say y^ ; then, as x varies, F(x, y) 
will become only where the RL. y = yi cuts the carve 
F{x,y) = ; but, as, and only as, x passes through a root x^ of 
F(x,yi) = 0, F(x,yi) changes sign. We may call the sides 
of the curve plus resp. minus. 

In the Eqs. of the curve and the tangent, (D) and (D'), let 
us replace 1 by -y resp. v^, so as to make them homogeneous in 
'^t y-i ^if Vi') '^5 '^1- They then take the forms 

kxx + h {xy -\- yx) + jyy -\-g(xv-i- vx) 

+f(y'" + "^y) + cvv = o, (E) 

kxj^x + 7i(x,y-{- y^x) -^-jyiy + g{x^v + v^x) 

-{-f{yiV-\-v,y)-^cv,v = 0. (E') 



LN^SIDE AJsB OUTSIDE. 83 

If, now, we proceed as sketched iu Art. 67 to find x^, we 
shall get a result of the form 

Ax, + Bi\ = ± vc; 

where A, B, C are functions of x', y\ v', k, h,j. g,f, c. 
Since V =- (/^^^"+%^ + r/i-0i^-i + (^>^^+/y'+c^'03% 

we readily see that C will be homogeneous of tenth degree in 
all the arguments : of fourth degree in x', y', v\ and of sixth 
degree in A.-, 7^ J, ^, /, c. [Consider that in replacing y, the 
parentheses ()i, ()2, ()3 will be squared, which squares will 
be again squared in completing the square (ina^i.^/i), as will the 
coefficients Z:, /i, etc.] Now, if (x',y') be on F{x,y ; x,y) = 0, 
the two values of a\ are equal; but then (7=0; hence, 
F{x',y' ; x',y') = makes (7=0; i.e., F{x' , y' ; x' , y') is 
a factor of (7. Xow, since x' enters F(x', y' ; x', y') in sec- 
ond degree, but enters (7 only in fourth degree, it follows that 
F(x', y' ; x', y') cannot appear in (7 in higher than second 
degree. If it appear in second degree, we may extract the 
second root, and wi^ite 

V"C = F(x', y' ; x', y') V^. 

Here B cannot contain x', y\ or v\ since each enters F in 
second degree, and each entered C in only fourth degree ; hence 
i? is a function of A:, /i, J,. ^, /, c only, and that of second degree. 
Hence, whether VC^be real or imaginary will depend only on 
the sign of i?, that is, only on 7j, 7i, J, /, (/, c, not at all on x\ 
2/'; that is, not at all on the position of the point (x',y') from 
which the tangents are drawn. Hence, tangents from all points 
will be either all real or all imaginary. This is so for any curve 
of second degree, hence for the special curve, circle. But now 
we know that some tangents to the circle (from points outside 
the circle) are real, while some (from points inside) are imagi- 
nary. Hence, F{x'^y' ; x',2/') cannot enter C in second 
degree, but only iu first degree. F{x\ y' ; x\ y') changes sign 
at every point of the curve F{x^ y ; .t, y) = 0, and no other 



84 CO-ORDINATE GEOMETRY. 

function of x and y does. Hence, C changes sign along the 
same curve, is plus for all points (x', y') on one side, minus for 
all points (x', y') on the other ; hence, tangents from all points 
on one side of the curve are real, from all points on the other 
side are imaginary. The side on which the tangents are real, 
we may call the outside ; the other, the inside. 

69. The Eq. -F(^\, yi; x, y)==0, of the tangent to 
F{x, y ', X, y) == is symmetric as to x and Xi, as to y and 2/i, 
a fact of highest import to the whole theory of curves of second 
degree. This import we shall now in part develop. 

Thus far, the point (oji, 2/1) has been taken on the curve ; the 
query is natural, What does F{xi, 2/1 ; x, y) = picture when 
(^11 2/1) ^s ^^t ^^ ^^® curve? To answer it, suppose tangents 
drawn from (.^i, 2/1) touching the curve at (cCs? 2/2) •> (^s? 2/3) • 
The Eq. of one is F(x2, 2/2 I ^? 2/) = 0- 

Since it goes through (cCi, 2/1)5 -^(^25 2/2 ; «^ii 2/i) = 0- But 
this Eq. also says that F(x, y ; ccj, y^) = goes through 
(^25 2/2) ; ^y liks reasoning, we show that it goes through 
(X3, 2/3) ; hence, 

F{oc, y ; x^, y^ = or F{x^, y^;oc,y) = 

is the ML. through the tangent points of tangents from (a?!, ?/i) 
to F{x,y;x,y)=:0. 

Such a EL. is named polar of the pole (%, 2/1) as to 
F{x,y; x,y) = 0. 

Since the equation of condition i^(%, 2/1 ; ^25 2/2) = ^ i^^y ^^ 
read either 

(i^i, ?/i) is on the polar of {x^, 2/2) ? i-e., on F(x, y ; y.^, y^ = 0, 
or (cc2, 2/2) "^'s 0^ t^^^ polar of (Xi, 2/1) ? i-^., on i^(a7i, 2/1 ; x,y) = 0, 

if one pole be on the polar of a second, the second is on the 
polar of the first; or, if one polar pass through the pole of a 
second, the second passes through the pole of the first. 



POLE AND POLAR. 85 

Two poles, each on the polar of the other, or two polars, 
each thi'ough the pole of the other, are called conjugate. 

Hence, if a point be on each of a system of polars, i.e., be 
their intersection, the poles of each polar will be on the polar of 
the point ; or, as a RL. turns around a point, its pole glides 
along the polar of the point ; or, as a point glides along a RL., 
its polar turns about the pole of the RL. 

If we convert the definition of the polar of a point, we shall 
get a definition of the pole of a RL. : as the intersection of the 
tangents to F{x^ y ; x,y) = through the intersection of the RL. 
and the curve. If, now, this RL. turn about a point, its pole 
will ghde along the polar of the point ; hence, the x>olar of a 
point is the locus of the intersection of the pair of tangents to the 
curve through the intersection with the curve of a RL. through 
the p)oint. 

70. These two definitions of polar are equivalent. The first 
yields a geometric construction only when the pole is outside the 
curve, for only then are the tangent points real. If the point 
be inside the curve, we may still draw through it two chords 
of the curve, and draw the two pau-s of tangents through their 
ends ; the RL. through the two intersections of the tangent 
pairs will then be the polar sought. When the pole falls on the 
curve, the Eq. shows the polar becomes a tangent through its 
pole as point of tangency. Hence, the tangent is to be thought 
as a p)olar through its own pole. 

It is carefully to note that the terms pole and p)olar are mean- 
ingless without reference, express or implied, to a curve of 
second degree, which we may call the referee. 

The notions of pole and polar are still deeper in wrapped in 
the notion of curve of second degree, as what follows may show. 

71. Be /xi : /X2 the ratio in which F{x., y \ cc, ?/) = cuts the 

1 i 

tract (it'i, 2/i) (a^s, 2/2) ; then the Cds. 

/^ia^2 + /^2«'l /^l2/2 + M22/l 

x= j -> y — i 

/^l + /^2 l^\-TH 



86 



co-ohdhstate geometry. 



of the section-point must satisfy F(x,y ; x^y) = 0. Substi- 
tuting herein and arranging terms, we get 

[N.B. Reason as in Art. 51 ; the result must be homoge- 
neous in /xi and /i.21 ^^^ symmetric as to indices 1, 2; the 
coefficient of jul^ resp. /xg^ we get b}^ supposing /xg resp. fx^ to be 
; the coefficient of /xi/xg naust be double and symmetric as to 
the indices 1, 2.] 

This quadratic yields two values of the ratio /xj : /xg, say r' 
and r". If (a.'i, 2/1) and (^^2,2/2) be conjugate (each on the other's 
polar), then i^(a^i, ?/i ; ct\,, ^/o) = 0, and then r'= — r"; i.e., 

the tract (.Ti, 2/1) (.^'2, 2/2) 5 from pole to polar, is cut by 
F(x, y ; X, y) = innerly and outerly in like ratio ; i.e., is cut 
harmonically. Hence, any tract from a pole to its polar is cut 
harmonically by the referee. 

Hence, once again, we may define polar thus : 

The locus of the harmonic conjugate to a fixed point, the other 
pair being section-points with the referee of chords through the 
point, is called the polar of the fixed point (as to the referee). 

Thus is justified the use o£ conjugate in Art. 69. 

7> 




If, now, there be given five points of the referee, we ma}^ 
construct it with the ruler only, thus : Through Pi and P2? -^3 



POLAE AS LOCUS OF A FOURTH HARMONIC. 



87 



and P4 draw secants meeting in J; on them find fourth har- 
monics * H^ j^' conjugate to J; HH^ is polar of I. Draw /P5 
cutting HH' at ^". Then is P^^ the fourth harmonic to P5, 
the other pair being 7, TT", a point of the Referee. As we may 
choose four out of five in five ways, and join each four by twos 
in three ways, we ma}' thus construct fifteen sixth-points. 
Recombiuing these twenty, we may go on to construct any 
number of points of the Referee. 



72. In Eq. (G) drop the subscript 2 ; then the roots r', r" 
are the ratios in which F{x^ y ; x^ y) = cuts the tract from 
(%, 2/1) to (x, y) . When these ratios are equal, the tract is cut 
by the curve in two consecutive points; i.e., the RL. through 
(a?!, 2/1) and (x, y) touches the curve ; but when the roots are 
equal, 

F(x^, 2/1 ; ^1^ 2/1) • ^(^5 y\^->y) = \f{x^, 2/1 ; ^^ y) V- 

In this Eq. (cc, ?/) is any point on a RL. through {x-^, y-^ tan- 
gent to F(x, y ; x,y) =0 ; hence, this Eq., being of second 
degree in {x, y) , pictures the pair of tangents through (a'l, y^ to 
F(x,y; x,y) = 0. 

The right member being a square, is always plus ; hence, the 
factors on the left are like-signed ; 
hence all points (cc, y) of a tan- 
gent lie on the same side as any 
one point (S/'i, 2/1) ; i-e., the tangent does not cut the curve; i.e., 
the curve is throughout convex or concave on the same side ; 
not like this figure. 




* This we may do by think- 
ing P^ P.^ a diagonal of a 
four-side cut by another di- 
agonal at I. Draw SP-^, SP2, 
SI; draw P^QR at Avill; 
draw RP^T; draw QT; it 
cuts P^P.^ at H, by Art. 49. 




88 



CO-OEDINATE GEOMETRY. 



Thus far all geometric representation has been purposely 
avoided, to show more clearly how the notions and properties 
of pole and polar all lie enfolded in the algebraic fact that in 
the Eq. of the tangent to the curve of second degree current 
Cds. and Cds. of the point of touch appear symmetrically. 




The figure illustrates the definitions given. Poles and polars 
are marked by the letters P and X, with corresponding indices. 
It is not necessary to know aught of the curve except that it is 
of second degree. 

We may now return to the special properties of the circle. 



73. By Art. 63, the section-points of x cos a-\-y sin a =p 
and x^ ■\-y^ = r^ are 

p cos a ± sin aVr^ —p^-) 



X-i 



2/1 = p sin a ip COS aV^'^ — i?^. 

The half -sums of these pairs are the Cds. of the mid-point of 
the intercepted chord : x^=p cos a, ?/,„ = p sin a. 

By eliminating j9 we get a relation holding between the Cds. 
of the mid-points of all chords having the same a, i.e., all 
II chords. Hence, 

ym''^m = tan a or 2/ = tan a • ic 



KORI^IALS. 89 

is the locus of the mid-points of a system of |! chords. Such 
a locus is called a diameter, and in this case is clearly a RL. 
through the centre _L to the chords. 

A RL. through any point (%, y-^ of a curye _L to the tangent 
at that point is named Normal to the curye at that point. 

The tangent to x^ + 2/' = ''^ ^t (a.\, y^ is xx-^ + yy^ = ?*^ ; 

^^^^' {x-x^)y^-{y-y^)x^ = 0, or xiy^x^-.y^, 

is normal to the circle x^ + ?/- = r at (a-j, 2/1) • 

The absolute in this Eq. is ; hence, 

Normals to a circle pass through (enyelop) a p)oint, the centre. 
Also, all normals are diameters of a ckcle. 

74. In the expression 



X - x{ + y-y{ — r, or xr ^ y- -^ 1 gx ■\- 2fy + c, 

either of which equivalents equated to is the X.F. of the 
rectang. Eq. of the cu'cle whose 

centi'e is (x\, ?/i), and radius ?', ^^^^i^^V) 

2 2 . 

x — Xr^-^-y — yx IS the squared 

distance of am* point {x^y^ from 

the centre ; and since the radius 

is J_ to the tangent at its end 




(Art. 73) , the difference x — % 
-\-y — y\— r is the squared tan- 
gent-length from (cc, ?/) to the 
circle x — x^^ -\-y — y{ — ^^=iO. 

So, too, is 0^- + ?/- + 2gx -\- 2fy + c the squared tangent-length 
from (.r, ?/) to the circle xr -\-y^ -{-2 gx + 2fy -|- c = 0. 

This squared tangent-length is called the power of the point 
{x, y) as to the circle. To find it, replace the current Cds. in 
the N.F. of the Eq. of the circle by the Cds. of the point. 

Like reasoning and results hold for oblique axes. The power 
of th.e origin (0, 0) is the absolute in the N.F. of the Eq. 

75. The centre being origin, the Eq. of the circle is xr -{- y^ 
= r^s however the axes be turned. Turn them till the X-axis 



90 



CO-OEDINATE GEOMETKY. 



passes through any assumed pole. 

^2 



Then is ?/i = 0, and the 



polar is xxi = r^ 



or 



X 



But this is a RL. J_ to X-axis, distant x from origin. 

Hence: (1) The circle being referee, tJie polar is _L to the 
radius through the pole; (2) the radius is the geometric mean 
of the distances of pole and polar from the centre. 




We may now trace the movement of pole and polar thus : 
When the pole P is at Pj, the polar L is the tangent Li ; when 
P falls on P^i L falls on L^ ; as P nears the centre, L retires 
to 00 ; as P passes through the centre, L passes through oo ; 
as P nears Pi', L nears ij' ; as P retires to go, i nears the 
centre ; as P passes through oo , L passes through the centre ; 
as P nears Pi, L nears L^. So long as P stays on a diameter, 
i stays II in all positions; i.e., L turns around its point oo, 
also P and L move counter. As P glides along a tangent, L 
turns about the point of tangency ; as P glides along any RL., 
L turns about the pole of that RL. ; as P glides around any 
circle concentric with the referee, radius a, L turns around a 



second concentric circle, radius 



second circle, L will turn around the first 



if P glide around this 
All these are special 



EXERCISES. 91 

cases of the general proposition : as the pole glides along 
(traces) any curve of nth degree^ the polar turns around (en- 
velops, enwraps) a curve of nth class; and conversely. Such 
pairs of curves are called reciprocal. 

N.B. A curve cut by a RL. in n points is of nth degree ; a 
curve touched by n RLs. through a point is of 7ith class. 
Curves of second degree and curves of second class are the 
sanae ; but in general curves do not rank alike in degree and 
class. See Art. 160. 

EXERCISES. 

1. Eind tangents to x'^ -{■ y^ — 6x — 14y — S —0 at the points whose 
X is 9. 

From the Eq. of the circle we find the corresponding value of y : 12, 2. 

The Eq. of the circle in the form F{x, y ; x, y) = is xx -^yy — S {x-\- x) 

— 7(_?/ + ^)— 3=0. Hence, the tangents are 

9x-\-12y-S{9-^x)-7 {12 + 2j)-B = 0, 
9x+ 2y-S{9 + x)-7 { 2+y)-S = 0; 

or, reduced, 6 a: + 5 3/ =114, 6x — by =4:4:. 

2. Similarly, find the tangents thus defined : 

x^-{-y^ — 4x + 22y + 2b = 0, x^ = S; 
(a: -5)2 +(^+8)2=113, t^ = 13. 

3. Eind the tangents to a:2+ 7/2 + lOx — 6 ?/ — 2 = II to 3/ = 2 r — 7. 
The Eq. of the circle may be Avritten (x + 5)2 ^ [y — 3)2 — 35 j or, if 

x' = X -\- b, y' = y — S, a:'2 4- 3/'2 = 36, The Eq. of the EL. becomes 
y' = 2x' — 20. A EL. || must be y'=2x' + b. This is tangent to 
ar'2 4- _y^2 _ 36 when, and only when, 36(l + 2-) = 62; i.e., when ?)=±6\/5. 
Therefore the tangents are y' — 2x'±6Vb; or, ?/ =2a:+ 13 ±6\/5. 

4. Draw tangents to x^ -{■ y'^= 58 inclined 60° to 4 a: — 3 y = 12. 

5. Through (x^ = 9, ^/^ > 0) on x'^ + y^-12x + 2y-]-S = draw RLs. 
inclined 45^ to the circle. 

Hint. The RLs. sought halve the angles between tangent and normal. 

6. Eind the angle between two tangents to a circle. 

7. Eind the power of (-11,-9) as to (r-3)2+ (?/ _ 7)2 = 25 ; of 
(4, 1) as to 4x2 + 4?/2 — 3 a: — ?/ — 7 = 0. 

8. Eind the circle tangent to y=Sx— 5, centre at origin. 

9. Find the tangents from (16, 11) to x'^ -\- y^= 169, and where they 
touch. 



92 CO-OEDINATE GEOMETRY. 

10. rind and draw the polars of : (11, 17) as to {x — 3)2 + {y + 5)2 = 81 ; 
(8,-5) as to x'^+ f + Ux+6i/ + 22=0; (-2,-7) as to x^ + y^ 
-18 a: + 2^ + 57=0. 

11. Find the polar of (x^, y^) as to the point (-circle) (a, 6). 
The Eq. of the point regarded as circle of vanishing radius is 

{x-a)2+ {^-6)2 = 0; 

.'. the polar of {x-^,y-^) is {x-^ — a){x — a) + {i/i — b){y — b) = 0. 

This RL. goes through {a,b) ± to the junction-line of (a:^, y{) and (a, b). 
Show that the polar of (11,3) as to (4, — 2) is 7 x+ By — 1^ = 0. 

12. Find the polar of (z^, ?/J as to a E,L. 

Regard the RL, as circle of infinite radius, and write its Eq. 

{x' ■{- y'){l + Jc) + 2 {A^-\- kA,) x+ 2 {B^ + hB,) y + C^ + kC,=0. 

Eor k = — 1 this circle passes over into a RL. The polar of {x-j^,yi) is, 
for k=-l, {A,~A,){x,+ x)+{B,-B,)(y,+ y)-^C,~C,^0, 
the RL. is 2(A^-A.^)x + 2{B^- B^) y -^ C^-G^ = 0. 

Hence, the polar is II to the RL., midway between it and the pole. 

Show that the polar of (8, 20) as to 5:r — 3j/ -j- 7 = is 5a: — 3?/ — 6 = 0. 

13. What is the pole : of y =mx + b as to x^ -\- y^ = r^ 1 

of Ax+Bi/-{-C=0 as to {x-a)^+{y-b)^=r^1 

14. Eind the pole {x2,y2) conjugate to (xj, 3/^) as to x^ ■}- y^= r^ and on 
the junction-line of (r^, y^) and the centre of the circle. 

Since {x^, y^), as conjugate to (x^^, y^), lies on the polar of {x^, y^), 
^1^2 + 2/13/2 = ^'^ ; since it lies on x:y = x^:y^, x^:y^ = x^: y^. 
Hence, x^ = r^x^ : {x^ -f y^), y^ = r'^y-^ : {x-^^ + y^). 

Hence, x^^ = r^x^ : (x^^ + y^^), y^ = r'^y^ : {x.^^ + y^^). 

Hence, {x^, y^) and [x^, y^) are called related as to x'^-\- y^= r^. 

15. Show that each of two related points is the pole of a RL. through 
the other II to the polar of the other. 



Systems of Circles. 

76. Be (7i=0, (72 = Eqs. of two circles in normal 
form. 

Then is Ci — XC2=0 the Eq. of a circle (by Art. 58) 
through the common 2^oints of Ci = and €2=0 (by 
Art. 30) . 



SYSTEMS OF CIRCLES. 



93 



Or, Ci — XC2=0 is the Eq, of a system or family of 
circles through two fixed points : the section-points of (7i = 
and (72 = ; A. is the parameter of the sj'stem, and ranges 
from — GO to + 00. 

Since G^ and C^ are the powers of any point as to (7i = 
respectively O2 = 0, and since X is clearly the ratio of 
these powers, we see that 

The ratio of the powers of any point on any circle of the 
system as to any two circles of the system is constant. The 
ratio is of course different for different circles or as to different 
pairs of circles. 

For A = l the terms of second degree vanish, the circle 
passes over into a RL., i.e., a circle of infinite radius. This 
RL. is always real, though the section-points be imaginary. 
Clearly the powers of its every point as to the two circles are 
equals since its Eq. is d— (72 = 0. Hence it might be 
named Equipotential Line or simply Power-Line of the circles 
(called Radical Axis by Gaultier, 1813). 

77. The power-lines of three circles taken in sets of two are 
(7i-(72 = 0, (72-(73 = 0, C3-(7i = 0; added, these 
Eqs. vanish identically; i.e., the three pyower-lines meet in a 
point — the power-centre of the three circles. 




Hence, to construct the power-line of two circles, (7i and (72, 
draw C^ and (7^ cutting (7i and €2- The power-centres of (72, 
(72, C^ and (7i, C^-, (74 fix two points on the power-line of Ci and 

o,. 



94 CO-ORDINATE GEOMETEY. 

78. Two points determine a RL. as common power-line of a 
system of circles: Oi — /\(72 = 0, through the points. The 
power-lines of each circle of such a system and a fixed circle C 
pass through a point. For the power-line of C and any circle 
C' of the system cuts the given power-line of the system say at 
7; which then is the power-centre of (7, C and any second 
circle C" of the system ; hence the power-line of C and C" also 
passes through I. 

EXERCISES. 

1. Find in co-ordinates the power-centre of (x — 7)^ + (?/ — 9)2=i 36, 
(a:+3)2+ (!/-2)2 = lG, (x-|-4)2+((/ + 5)2=z9, and draw the figure. 

2. Show tliat the power-line of two circles is _L to the junction-line of 
their centres (or centime-line, as it may be called). 

79. The form Ci — A(72 = is not convenient for study- 
ing a system of circles. The power-line and junction-line of 
centres, being _L, naturally suggest themselves as axes. The 
latter being taken for X-axis, the term in y falls away ; also, 
for x= the values of y are equal and unlike-signed for 
all the circles ; hence the parameter A can enter only the term 
in cc, and we can write the Eq. of the system, 

Here A is the changing distance of the centre from the origin ; 
8 is the fixed distance to the section-points from the origin ; 
these points are real or imaginary, according as 8^ is — or -f-. 

80. The Eq. of the system of polars of any point (x^, 2/1) as 
to this system of circles is x^x-i-yiy — X(xi-j-x) -\-S^ = ; 
A appears in first degree only, hence this system of polars pass 
through a point, the section of XiX -{- y^y -\-d^ = and 
Xi-\-x = 0. 

In general, then, the polar of a point changes with the circle 
of the system, turning about a point ; but if the two RLs. which 
fix this point, x^x -\- yy^ + ^^ = , Xi-\-x = 0, be the same 



OKTHOGOXAL CIUCLES. 95 

RL., then are all RLs. of the system x^x-\-y^y — X{x^-[-x) 
-f S^ = the same RL. 

This is so when, and only when, y^= 0, ^1= ± S, for 
otherwise the two Eqs. are not the same. Hence each of these 
two and only Vhqsq points (8, 0), ( —8, 0) has the same polar as 
to all circles of the system, namely, a RL. through the other 
± to the line of centres. 

These points are real or imaginary according as 8^ is + or — , 
i.e., according as the section-points of the circles are imaginary 
or real. Writing the Eq. of the system thus, 

fj^{x-xy = x'-h\ 

we see if 8" be +, and so the above critical points real, then 
the circle is imaginary, for ever}' A < 8. For A very large the 
centre retires toward x along the X-axis, the circle flattens 
toward the Y'-axis ; as X nears 8, the centre nears the critical 
point (8, 0) , the circle shrinks toward and around that point ; 
and as X equals 8, the circle vanishes in that point. Hence the 
critical points (8,0), (—8,0) are themselves circles of the 
S3'stem, point-circles, and are hence named by Poncelet limiting 
points of the system. 

81. The powers of any point of the common power-line as to 
these circles, i.e., the squared tangent-lengths from any point 
of the power-line to the circles, are all equal ; i.e., the ends of 
all such tangent-tracts, the points of tangenc}', lie on a circle 
with centre on the common power-line. The radii of this circle, 
as tangent to the other circles, are _L to the radii of those circles ; 
i.e., each circle tvith centre a point of the poiver-line and squared 
radius the power of that point cuts orthogoncdly the ivhole system 
of circles. 

As the limiting points are circles of the system, each orthogo- 
ncd circle passes through the limiting points. Hence the Eq. of 
the system of orthogonal circles is 

x" + 2/2 _ 2 A?/ - 8- = = x2 + (?/ - X)2 - A^ - Z\ 

Deduce this directly as the Eq. of the orthogonal system. 



96 



CO-OEDINATE GEOMETRY. 



Note that these two mutually orthogonal systems are comple- 
mentary : in the one S^ is + , in the other — ; the power-line of 
one is the centre-line of the oth.er ; the section-points of one 
are the limiting points of the other ; of the one the section- 
points are imaginary, the limiting points real, — of the other, 
vice versa. 




To construct this double system. Draw any number of cir- 
cles through two points. To any one draw any number of 
tangents. About the points where these cut the RL. of the 
two fixed points describe circles with the tangent-tracts as 
radii. Do not fail to carry out this construction. 



EXERCISES. 

1. When are C=0 and C^^O ±? 

The squared distance between their centres must equal the sum of 
their squared radii; i.e., 



EXERCISES. 97 

or, 2gg^+2ff^-c-c^ = 0. (A) 

Hence, if (7=0 cut both (7^ = and Cg = at right angles, we 

have 

25'i^ + 2/i/-c-Ci = and 2^72 5' + 2/;/-c - c^^ 0. 

From these we may express any two of the three symbols g, f, c 
through the other linearly ; substituting in (7=0 we get an Eq. of a 
circle containing one parameter linearly ; hence all circles cutting two 
circles orthogonally form a system through two points, i.e., with common 
power-line, as already proved. 

If (7=0 cut three circles orthogonally, then we have three Eqs. of 
the form (A), which with (7=0 give, through elimination of g,f, c, as 
the equivalent of (7=0: 

= 0. 



X^ 4- y^ 


— X 


-y 


1 


Ci 


9i 


A 


1 


^2 


9-2 


A 


1 


C3 


93 


fz 


1 



Geometrically this is clearly a circle about the power-centre, its squared 
radius the power of that centre. 

2. The circle orthogonal to C^ = 0, C^ = 0, C^ = is also orthog- 
onal to AjCi -j- A2C2 + ^3^3 = 0- Use condition (A). 

3. The polar of one end of a diameter of a circle as to any orthogonal 
circle passes through the other end. 

Be x"^ -\- y^ = r'^ the given circle ; then by (A) an orthogonal circle 
is x^ + 3/^ -f 2 ^a: + 2fy + r^ = 0. As to this the polar of (r, 0) is 

rx -f ^(r + x) -f- r^ = 0, 
which always goes through (— r, 0). 

4. Powers of points of one circle as to another vary as their distances 
from the power-line. 

5. To find the angle a^ under which (7=0 and (7^ = intersect, 
a^ equals the angle between the radii to a section-point. If r, r^ be the 
radii, d the distance between the centres, then 

2 ?Tj cos Oj = r^ + r-^ — <P. 
If we hold (7^ = fixed and a^ constant, then since d} — r^ is the squared 
tangent-length to C^ = from the centre of C = 0, we have this 
relation between the Cds. of the centres of all circles, (7=0, which cut 
C-^ = 0, under the angle a^ : 

r'^ — 2 r^r cos Oj = C^. (H) 



98 CO-ORDINATE GEOMETEY. 

Since x and y enter C^ this Eq. contains tliree arbitraries, x, y, r. Impos- 
ing the further condition that C= cut C.^^O under 0.2, we get a 
second Eq. : ?-^ — 2 r^ cos a^ = C^. These two Eqs. do not yet fix the 
centre [x, ij) and the radius ?' of (7^ = 0; but they determine its family. 
Eor it will cut any circle of the system C^ — A (72 = under a constant 
angle. We have, namely, 

2 _ n , »'i cos a^ — Xi\ cos a^_ C| — xC^ 

\' 1-A ~ 1-A 

which declares that the varying circle (7=0 cuts the circle 

C'= ^i~^^^ =rO, 

1-A 

radius r', under an angle 7 such that 

r' cos 7 = (r^ cos a^ — Xr^ cos a^): {1 —X). 
We may express ?■' through the constants of (7^ and C.^ thus : 

r'2 = {(1 - A) (r,2 _ Ar^^) - Xd^ : (1 - A)^ 
Hence 7 is determined univocally, i.e., is constant. 

If we assign 7 at will and substitute the value of r' , we get a quadratic 
for determining A ; i.e., there are tiio circles of the system G-^ — xC.^ = 
which the varying circle C=0 cuts under any given angle. As a 
special case, for 7=0, cos 7=1, we see that there are two circles of 
the system Avhich the varying circle always touches, on which it rolls. 

6. Through the section-points of x^ + ?/^ + 4 j: — 14 ^ — 68 = and 
a:^ 4- ?/2 _ (3 ^ _ 22 ?/ -f 30 = draw a circle tangent to X-axis. 

7. Under what angle do x'^ -{■ y'^ = IQ and (x — 5)^ + 3/^ = 9 
intersect ? 

8. Find the power-centre of 

a:2 + ^2 _ 2 :c + 6 _?/ - 15 = 0, 

a:2 + ?/2 -f 14 a: 4- 12 v/ -f 81 = 0, and the point (3, -7). 

9. rind a circle through (— 5, —4) and cutting orthogonally 

10. Eind a circle through (- 4, 3), (— 2, — 3) cutting x"- ^r y'^ — Q x — 
7 = orthogonally. 

11. Show that the point-circles cut a diameter of every circle of the 
system harmonically. 

12. AVh at circle of x^ -f- / - 2 A:c -f S^ = cuts {x -a)^+ {y -hf = 7^ 
orthogonally ? 



CENTRES AND AXES OF SIMILITUDE. 99 

13. It has been shown (Art. 80) that the polars of a pomt as to a sys- 
tem of circles pass through a point, — the centre of the polar family. 
These two points are called poles harmonic as to the system of circles. 
Show that the power-line halves the distance between any two harmonic 
poles, and that the circles cut the junction-line of the poles in an involu- 
tion of points. 

14. Find a circle whose power-lines with two given circles go through 
their centres. 

15. Show that the centres of all such circles lie on a RL. II to the 
power-line of the two circles, named secondarij power-line. 

16. Show that the circles halving the circumferences of two given 
circles form a system, and find the power-line. 

17. Show that the secondary pov»'er-lines of three circles go through a 
point. 

18. Find a circle halving the circumferences of the three circles : 

x'^^ y^-2x — Q = 0, .r2-l-/ + 3x — 9 = 0, x'-^ -f ^2 _ g^/ _ lOx + 18 = 0. 



Centres and Axes of Similitude. 

82. Congruent figures are alike in shape and size, differ 
only in place ; they may be thought fitted one on another. 

Similar figures are alike in shape, but not in size. Such 
figures may be supposed made thus : 

From any point clraiu rays in any directions ; on each ray take 
two tracts in a fixed ratio; the ends form two similar figures. 
The fixed point is called centre of similitude; the fixed ratio, 
rat io of si mil itude . 

Clearly the one figure may be thought as the other swollen 
or shrunk in like measure throughout. 

By pushing or turning either figure the shape is not changed. 
When simply pushed, or when turned through a flat angle, cor- 
responding tracts in the two figures keep H, and the figures 
keep similarly placed. 

By the above construction of similar figures, space is doubled 
in thought : the space of the one figure, and the space of the 
other. To each point in one space corresponds a point in the 



100 



CO-OEDINATE GEOMETHY. 



other. If these spaces be now thought pushed or turned out 
from each other, corresponding points remain centres of simi- 
larity for the two figures and spaces. For, by similar triangles, 
corresponding tracts, tracts between corresponding points in 
pairs, are proportional and include equal angles. 

In what follows we shall keep the original construction, with- 
out pushing or turning, unless through a flat angle. 




83. Clearly, tangents at corresponding points, being drawn 
through pairs of corresponding points, are H. 

To find the figure c' similar to a circle c, radius r, ratio of 
similitude r:r'. Take any point as a centre of similitude, 
lay off OC so that r:r'= OC: OC. Let P' correspond to P, 
then OF: OP' = r:r'== OC: OC = CP: C'P' = r :r'. 

Since CP is constant, so is C'P' ; i.e., c' is a circle, radius r' ; 
i.e., a figure similar to a circle is a circle. 




Conversely, all circles are similar. For be c and c' any two 
circles, radii r and r' ; cut the tract between their circles innerly 
respectivel}' outerly in the ratio r-.r'-, then, by the above, the 
figure similar to c is a circle with centre C, radius r' ; i.e., it is 
the circle c'. 



AXES OF SIMILITUDE. 101 

CoE. 1. Any two circles in a plane have two centres of simil- 
itude : inner and outer, cutting the tract between their centres : 
innerly and outerly, in the ratio of their radii. This j)roperty is 
peculiar to the circle, since it alone of plane figures, being 
homogeneous, may be turned around in itself. 

Con. 2. The two pairs of common tangents to two circles 
cross in the inner resp. outer centre of similitude. Or thus : 

A tangent to one of two circles through a centre of simili- 
tude is tangent to the other ; for the radii to the points of touch 
are H. 

84. Be C^= (x — x^y-j-(y — y^y — rj = any circle , and 
let /(CiCg) resp. E{CiC^ denote the inner resp. outer centre 
of similitude of C^ and CV Since I(CiC2) resp. E^CiC^) cuts 
the tract between the centres in the ratio of the radii, its Cds. 
are 

r^X2-\-nx^ ^12/2 + ^22/1 

^ =- ; 1 V = i ' 

^1 + ^2 n + r2 

,, f\X2 '^2^1 ,, ^l2/2 — '^'22/1 

resp. x"= 1 y"= • 

'1 — '2 '1 — ^2 

(x', y') resp. {x"^ y") is the pole of the chord of contact T'T' 
resp.. T"T" of the inner resp. outer common tangents. Hence 
substituting for x', y' resp. x", y" in the Eq. of the polar, we 
get as the Eqs. of these four chords, after easy reduction : 

{X2 - x^) {x — x{) + (y.2 — 2/1) (2/ - 2/1) = ^1(^1 =F ^2) , 

resp. (xi - X2) {x - X2) + (2/1 - 2/2) (2/ - 2/2) = ^2 (^'2 =F ^1) • 

The centre-line is (?/2 — 2/1) {x — x^) — (X2 — x^) (y —y{) = 0, 
whence we see the chords of contact are JL to the centre-line. 

85. Three circles, C^ C2, C3, combining three ways in sets of 
two, have six centres of similitude : three inner, three outer. 
Form the Eq. of the RL. through E{C^a^ and ^(aCg), multi- 
ply it by (ri — i^) (^2 — r^) , and divide it by 9-2 '■> the result is 



102 



CO-OEDINATE GEOMETKY. 



7\ 7-2 Ts 


X — 


2/1 2/2 2/3 

1 1 1 





n 7-2 Vs 


2/ + 


ri rs Vs 


1 2 3 




2/i 2/2 2/3 


1 1 1 




'^l ^2 "^3 



= 0. 



(H') 



To find the RL. through ^(OgOg) and E{C^C^) permute the 
indices ; this will not change H\ it being symmetric as to the 
indices ; i.e., the RLs. are the same ; i.e., the three outer centres 
of similitude, E{CiC2), ^^(CaOg), E{C^Ci) lie on a RL. By 
changing the signs of the r's properl}^ we show that 

E{c,a,), i{ac,), i{C,c,)', 

E(aC,), 1(0,0,), 1(0,0,); 
resp. E{0,0{), 1(0,0,), 1(0,0,) 

lie on a RL. T7ie.se four RLs. are named axes of similitude of 

the three circles : one outer, three i^mer. 

Cor. If two circles touch innerly resp. outerh', the point of 
touch is an outer resp. inner centre of smiilitude of the two ; 
hence, from the above, if one circle touch two, the junction- 
line of the points of touch passes through a centre of simiUtude 
of the two : outer resp. inner according as the circles are 
touched alike (both outerly or both innerly) resp. not alike 
(one innerlj', one outerly). 



EXERCISE. 

Fmd the centres and axes of 

x2 + ^2 ^ 16, {x - 5)2 + / = 81, [x-yY-\- [ij - 10)2 = 4. 
Draw the figure. 

86. If a A^arying circle cut the circle Ci = under the 
angle a, by Art. 81, its radius R and its centre-Cds. satisfy the 
Eq. R^ — 2 r,R cos a = O,. Treating R and a, orj what is the 
same, R'^ and jRcosa as parameters, we may impose two more 
such conditions : 

R"^ — 2r,RcoQa= 0„ R^ — 2 r,R cos a = O,; 

eliminating the parameters from these three Eqs., we get a 
relation between the centre-Cds. and constants, i.e., we find 



THE TACTION PROBLEM. 108 

the locus of the centre of the ckcle cutting the three cu"cle3 
(7i = 0, (72=0, (73 = under the same angle a. So we get 

0\ - rs) {C, - a) - (r, - n) (C, - Q = 0, 
a RL. through the intersection of Ci— €2 = and C^— Cq = 0, 
i.e., through the poioer-centre of (7i = 0, C.2 = 0, C2, = 0. 

Writing for C-^— Co-, Ci— Co their vahies, we find the 
coefucients of x resp. y are the coefficients of y resp. —x'vn 
the Eq. of an axis of similitude ; i.e., the EL. is J_ to such an 
axis, and in fact the outer one. For thus far we have taken a 
as the inner angle between the circles, i.e., the angle between 
the radii to a section-point. To take a as the outer angle in 
case of either of the circles, it suffices : to change the sign of 
cos a, since the inner and the outer angle are supplementary and 
cosines of supplementary angles are equal and unlike-signed ; 
or to change the sign of r^, r^ resp. 73. Taking all the angles 
as outer changes all the r's, which does not affect the Eq. of 
the RL. ; changing one of the 7''s and leaving two unchanged is 
clearly tantamount to changing the two and leaving the one 
unchanged ; this can be done in three ways ; so we get three 
other RLs. through the power-centre JL each to an inner axis of 
similitucle. Hence the whole locus of the centre of a circle 
cutting three circles under the same (varying) angle is a pencil 
of four RLs. through the power-centre. _L each to an axis of 
similitude. 

87. If two circles touch innerly, their inner angle is ; if 
outerly, it is 180°. Hence, the J_ from the power-centre on the 
outer axis of similitude contains the centres of two circles : one 
touching the three circles all innerly ; the other, outerly ; the 
r's are all -j- or all — . Changing the sign of one r we get a 
J_ on an inner axis of similitude containing the centres of two 
circles : one touching two circles innerly, the third outerly ; the 
other touching the two outerly, the third innerly. Changing 
the sisfn of each r in turn we find in all eio-ht circles touching: 
each of three circles ; a pair of centres lie on each _L through 
the power-centre on an axis of similitude. 



104 



CO-OKDINATE GEOMETEY. 



We might now determine another hne on which the centre of 
a tangent circle must lie, by eliminating R between two Eqs. of 
condition, as C^ — Br + 2jRri, C2 = i?^ -f- 2i?r2. But the line 
would not turn out to be a RL. or a circle, hence would not 
admit of elementary construction : with compasses and ruler. 
But the doctrines of poles and polars, power-centres and 
power-lines, centres and axes of similitude, enable us to solve 
the general Taction-Froblem by use of the ruler alone. 

88. Suppose the circle resp. 0' touches the given circles 
Gil ^2-) Cq outerl}^ resp. innerly. 

(1) Then by Art. 85, Cor., the chords of contact 7\r/, 
T2T2', T^T^ go through P, the inner centre of similitude of 
and 0'. 




(2) Hence, FT,: PS,'=PS,: PT,', PT,- PT^ = PS,' PS^. 
But PTi . PS,- PT,'- PSi'= (f • q'\ (f and g'" being powers of 
Pas to and 0'. Hence, PT,- PT^=qq' (a constant for 
all directions) = PT. • PT<^^ PT^ • PT^ ; hence Pis the power- 
centre of (7i, C2, C3. 

(N.B. T^ and Tj' are 2iXL\i-correspondent points of and 0'.) 



THE TACTION PROBLEM. 105 

(3) Since Oi and C2 touch resp. 0' each outerly resp. 
inneiiy, as in (1), the contact-chords T^Ts resp. Tj'Tg' go 
through the same outer centre of similitude of Ci and C^, say 
K. Hence, as in (2), 

XT, . KT, = XT,' . XT2' ; 

i.e., the powers of X as to and 0' are equal; i.e., X is on 
the power-line of and 0'. So too likewise are the outer 
centres of similitude of C2 and Cg, Cg and (7i ; i.e., the outer 
axis of similitude of , O,, Og is the power-line of and 0'. 

(4) The power-lines of and C^ resp. 0' and d, i.e., the 
tangents at T^ resp. 2\', meet say at Xj, which is then the pole 
of the contact-chord T^T^' as to Ci ; but b}^ Art. 77 the power- 
line of and 0' also goes through Xi, the power-centre of 
0, 0', Oil i.e., through the pole of chord T^Ti' as to (7i ; 
hence, by Art. 69 the chord T^T^' goes through the pole of the 
power-line as to Ci- 

Likewise the chord T2T2 resp. T^T^' goes through the pole of 
the same power-line as to C2 resp. C.. 

If, of the two circles and 0', touch Ci and C2 outerly 
resp. innerly and Q innerly resp. outerly, but 0' touch (7i and 
C, innerly resp. outerly and C^ outerly resp. innerly, then the 
power-line of and 0' passes through the outer centre of simil- 
itude of Ci and (72, through the inner centres of similitude of 
C2 and Cs, Cg and C^ ; i.e., it is the inner axis of similitude 
^3/1/2 ; the other relations hold unchanged. Hence the follow- 
ing rule : 

Determine the power-centre and axes of similitude of the 
three given circles ; determine the pole of each axis as to each 
circle : each three RLs. through the power-centre and the three 
poles of each axis cut the three circles in contact-points of two 
tangent circles. 

Remark. This classic problem, in which the geometry of the 
circle seems to culminate, was proposed and solved by Apollo- 
nius of Pergse (b.c. 220). His solution was lost, but was 



106 



CO-OEDINATE GEOMETBY. 



restored by Vieta (tl603). The first solution of the analogous' 
problem for space : to find a sphere touching 4 given spheres, 
was given by Fermat (11665). Both solutions were indirect, 
reducing the problem to simpler and simpler problems. Gaul- 
tier (1813) and Gergonne (1814) first gave direct solutions of 
the first problem. In the above rule replace 3, circle, axis by 
4, sphere, plane, to solve the problem for space. Note care- 
fully on what the solution turns : on determining the chords of 
contact in the given circles (spheres) by two points : the power- 
centre and a pole of an axis (plane) of similitude. ■ 



Circular Loci. 

89. 1. Given any A cut by a transversal through a fixed point of the 
base; through the fixed point and the intersections of the transversal with 

each side and the adjacent vertex 
-n at the base are drawn circles ; find 

tlie locus of their intersection. 

Be ABA' the A. Take the fixed 
point as origin, the base as one 
rectangular axis, say -X-axis. Be 
OA=a, OA' = a'; then the sides 
AB, A'B are 

y = c{x — a), 
y = c'{x — a'); 

the transversal is y ~sx ; it cuts 
the sides at 

'^'^ or C, C. 




J,.r 



c — s c—sj \c' — s c — s 
The circles through 0, A, Cresp. 0, A', C are 



x^ ■}- y^ — ^^ 



a(cs -lI) r. 

c —s 



resp. x'^ + y^ 



a'x i ■ '- y 



Eliminating s, we get the locus of their intersection, 

{c{x^ + ^^ — o.^) — Ciy} '' {^^ ■\- y^ ■ — CL3C-\- cay\ 
= {c^{x^ -h y^ — «'^) — «'i/} • {•^^ + 2/^ — <^'-3^ + c'a'?/} ; 
or, {c — c') {{x^ + y'^)^— {a + a')x{x^ -\- y^) + aa^{x^-\- y^)^ 



or. 



CIBCUI.AII LOCI. 107 

{l + cc'){a-a') ,1 



r'+f) } x^ + y^-(a + a')x- ^ i + cc j(o-a ; ^ ^ ^^/ L^q 
i c— d ) 




This Eq. is the product of two : the first, of a point-circle, the origin ; 
the second, of the circle circumscribing the A. 

2. A right angle turns about a fixed point ; find the locus of the foot of 
the _L from the vertex on the chord of the intercept of its sides on a fixed 
circle. 

The Eq. of the circle, referred to rectang. axes through the centre, is 

:r2_^i/2=r2. (1) 

Take the diameter through the fixed point P 
as ^-axis. The chord is 

y=L sx -{■}). (2) 

Eliminate y between (1) and (2), whence, 

(l + s2)^2_^2s6a: + 62-r2=0. (3) 

The roots of this Eq., x^, x^, are the x's of A, 
B; the ?/'s are sx^ + h, sx^ +6. If OP = c, the direction-coefficients of the 
angle's sides PA, PB are 

sxj -j- h sx^ -f & . 
, , 

since the angle is right, their product is — 1 ; which, cleared of fractions, 
gives 

(1 + S-) Xj^X,^ + (a/j - C) (Ti -f X^) + i2 ^ c2 :=: 0. 

From (3) we have 

x^x,= {b''-r''):{l + s^), x^ + x^=-2sb:{l + s^)' 
:. (l-^s^){c2-r2)-f 2 6(sc-f 6) = 0, (4) 

an Eq. of condition between the parameters s and 6. 
The ± from P on the chord AB is 

3/ = --(^-c). (5) 

Eliminating s and h, by means of (2), (4), (5), we get the locus of the 
intersection of (2) and (5), the locus sought: 

I ?/2-|-(x-c)2| . |x2-f y,''-c:r + -^^^::^| = 0. 

This breaks up into the Eq. of the point-circle P, and the circle about 

the mid-point of OP, radius -* /jf _ _£!. 

\ 9 A 



108 CO-OEDINATE GEOMETRY. 

Clearly P is not part of the locus sought ; how, then, does it appear as 
part in the result? The pair of values, x—c, y = 0, satisfies (5) for 
all values, real and imaginary, of s and b; then, from (2), 0—sc^b; 
hence, from (4), for c r, (1 -f s^) = 0, s = ± L 

Now the problem as proposed implied only real values of s and b, but 
the analytic statement held not only for real, but also for imaginary, 
values ; i.e., for the problem in question and for more ; accordingly, the 
result yields the locus sought, for real values of s and b, and another locus 
not sought, for imaginary values of s and b. 

3. Find the locus of the foot of the ± from the centre on the chord. 

4. Find the locus of the intersection of tangents at the ends of the 
chord. 

Be (a;', ?/^) the intersection; then x' x -\- y' y = r'^ is the chord. Combine 
this with x^ 4- y/^ = r^ ; the pairs of roots so obtained (x^, y-^), {x.^, y^) picture 
the ends A, B of the chord ; the coefficients of direction of PA, PB are 



Vi Ih 



_, and their product is 



Hence, [x''^ + y''^){r^ — c^) + 2r2cx' — 2r*=: 0; 

or, dropping the primes, 



I ^ "^ ,-2 _ c'i ) 



^^ ^,_r^( 2r^-c^) 



If R be the radius of this circle, d the distance between the centres, 

then {m - d'^Y^ 2 r' {B" + d""), 

the condition that a quadrilateral inscribed in one circle may be circum- 
scribed about another, any tangent to the inner being taken as a side. 

5. Find the locus of a point the feet of perpendiculars from which, on 
the sides of a A, lie on a RL, 

Be x cos a^ + ?/ sin a^ — jo^ = = iVj, N^ = 0, iVg = the sides of the 
A; (^ij ^i), (•^25^2)5 (^3' J/3) the feet of the ±s ; {x,y) the point. Then are 
iVj, N^, iVg the lengths of these ±s. Their projections on the axes are 

X — x-^— N-^ cos a^, y — ?/j = iVj sin a^, 
and BO with indices 2, 3 ; hence, 

xu = x — Nu cos ah, yu—y— Ntc sin a^ 
for h ^ 1, 2, 3. 



CIKGULAIi LOCI. 109 

The feet {x-^, y-^), [x^, y^J, {x.^, y^) lie on a EL. when, and only when, 

or, after reduction and substitution, 

iV^iVg sin ( «! — a.2) + ^^2^^$ sin ( 03 — ag) + iVgiV";^ sin (wg — a^) = 0. ( 1 ) 

A\, iVs' -^3 ^^^ linear in x, y ; hence, (1) is quadratic in x, y. The first 
term yields as coefficients of x^, y"^, resp. xy, 

cos oj^ cos a2 sin (a^ — 02), 

sin a^ sin a^ sin [a-^^ — a^), 

resp. sin (a^ + 02) sin (a^ — 02). 

The difference of the first two is 

cos (cj^ + 0.2) s^^ ('^i — "2)5 
or, 2^ (sin 2 Cj^ — sin 2 02). 

The third is —J (cos 2 a^ — cos 2 02). 

Permuting the indices to get the contributions of the other terms, and 
summing, we see that the difference of the coefficients of x"^ and i"^, as 
well as the coefficient of xy, vanishes; i.e., the locus is a circle; the Eq. 
is also satisfied by putting any two of the iV's=0; i.e., the circle goes 
through the vertices. 

. Hence, the feet of ±s on the sides of a A from any point of the circum- 
scribed circle, and from no other, lie on a EL. 

This problem deserves further notice. Suppose the ELs. N-^—0, 
N^ = 0, N.^ — tangent to a circle 0, radius r, centre at origin; then, 
p^=zp^=p^ = r. Developing (1), we have 

M {x'^+y'')-Px-Qy + F=0, 

where M = sin {a-j^ — a.^) sin (03 — a.^) sin {a^— a^) 

If R be the radius of this circle, D the distance between the centres, 



M 



Z)2= (P'^+ Q2):41f2^ 
F 



whence, D^ —R"^ — 

M 

If A-^, J-2, A^ be the angles of the A, then 

02 — a^ = TT — J.3, 03 — 02 = TT—J.^, a^ — a^ — Tr—A,^; 

hence, if S= area of A, M— — sin^^ . sin ^2 • sin^^g = — S:2B^', 

and from (2), i^ = r^ (sin J-i+ sinJ.2+ sin ^3). 



110 



CO-ORDINATE GEOMETRY. 



If Sp s.,, S3 be sides of the A, 

2 S=r (sj + s^ + S3) — 2 Rr (sin A^ + sin A.2 + sin A^) ; 

whence, F = i'S : B, and hence, F : il/= — 2 i?r ; 

.-. Z>'^ = i?2 _ 2 i?r. (I) 

Such, then, is tlie relation connecting the radii and distance heticeen the 
centres of the inscribed and circumscribed circles of a A. Now, holding the 
circles fixed, let us see how we can vary the A. 

Taking the one centre as origin, the other on the J^-axis, we have 



(2 = 0, R' = 



4:M-^ M' 4:M' 

By virtue of (I), these three relations are satisfied if these two are : 
Q=0, F:M=-2Rr. 

Choosing one of the angles a^, a^, a.^ at pleasure, we can still find values 
of the other two to satisfy these two equations. 

Hence, When the relation TJ^ =^W — 2 Rr holds heticeen the radii of tico 
circles and the distance between their centres, a A can he drawn in the one ahout 
the other, the direction of one side being taken at pleasure. 

Theorems analogous to these two for triangles and quadrilaterals liold 
for polygons generally. 

We got (1) by imposing the condition that the feet of the ±s lie on a 
EL. Let us see how it expresses this condition. Suppose the origin 

inside of the A and o.^ < CI2 ^ °3- ^^ 
P{x, y) any point within the A, 
and join the feet of the ±s from it 
on the sides of tlie A, to form a A 
F^, F.„ F.. Then N^, N^, iVg are 
the lengths of these ±s, all have 
the same sign — , and are like- 
signed with the ±s from the origin 
inclined a^, o^_, a^, as both P and 
the origin are within the A. Hence 
the -terms of the left side of (1) 
are in order the double areas of 
the As F^PF,, F.PF^, F.,FF„ the 
whole left side is tlie double area 
of F,F,F,. 
If P'{x, ?/) be without the A, then one of the ±s, say N^, becomes +, 
and the left side of (1) is the difference between the double area of the A 
F.JP'F./ and the sum of the double areas of F^' P' F,/ and F.JP'F/, i.e., 
again, the whole left side is the double area of F^' F.J FJ. Now this double 




CIECULAE LOCI. Ill 

area is when, and only when, F-^, F,^, F.^ are on a RL. Accordingly wo 
may generalize our problem by requiring that the area of the A F^F.^F.^ 
be not but some constant ± c'^. Then the left side of (1) is this area 
doubled; also since (1) is the Eq. of a circle whose radius we call R, the 
left side can be written K{d'^ —R^), where K is constant and d^ stands for 
the general expression, x'^ + y- -{- 2 gx + 2///. Hence 

iVjiV^g si^ (^2 — '^'i) + ^2^^3 sin {a^ — a^) + ^3-^^ sin {a^ — a.) 
= ±2c'^K{cP-m) 

is the locus of a point the feet of _Ls from vdiich on the sides iV^ = 0, 
N^ = 0, iVg = of a A are vertices of a A of constant area c^, -f or — . 
The locus is two circles concentric to the circle circumscribing 'the given 
A. The outer is always real ; the inner, only when c- < KB^. The given 
area c^ changes sign for d—B, i.e., as P goes through the circle. 

6. Find the locus of a point, the sum of whose squared distances from 
n points, multiplied resp. by given constants, shall be a given constant. 

7. Find the locus of the centre of a circle seen from two given points 
under given angles. 

8. Find the locus of the centre of a circle that cuts two given circles 
at ends of diameters of each. 

9. From a fixed point P are drawn tangents to a system of circles 
through two fixed points ; ' find the locus of the intersection of tlie chord of 
contact with the diameter through P. 

10. Be 1 2 3 1^ 2' 3' a regular hexagon ; draw 1 3, 1 3', also any RL. 
through the centre, cutting 1 3, 1 3' at 4, 4' ; find the locus of the intersec- 
tion of 2 4 and 2' 4'. 

11. Find the locus of a point whose polars as to three given circles 
meet in a point. 

12. A constant angle turns about its vertex fixed on the bisector of a 
fixed angle; find where the J_ from the vertex on the junction-line of the 
intersection of the sides of the angles meets it. 

13. Find the locus of a point from Avhich two given circles seem of 
like size. 

14. Find the locus of a point whence two consecutive tracts LM, MN 
of a EL. seem of like size. 

15. Find the locus of the mass-centre of a A inscribed in a given circle, 
on a given chord as base. 



112 CO-ORDIKATE GEOMETRY. 

16. Under the same conditions as in (15) find the locus of the ortho- 
centre and of the centre of sides of the A. 

17. Of two related poles, as to a given circle, one glides along a RL. ; 
how does the other glide ? 

18. How does one of two related poles as to a:'^ ■\- y^—r^ glide when 
the other glides along {x — a)"- ^ y"^ — W' ^ 

19. rind the locus of the mid-point of a chord of a given circle, which 
subtends a right angle at a given point. 

20. Find the locus of the foot of the ± from the origin on a chord of 
a given circle, which subtends a right angle at the origin. 

21. Two variable circles touch each other and two fixed circles ; find 
the locus of their point of touch. 

22. Find the locus of a point whose polars as to three fixed circles 
meet in a point. 



GEKEEAL PEOPEKTIES OF CONICS, 



113 



CHAPTER IV. 



GENERAL PROPERTIES OF CONICS. 

90. The general Eq. of curves of second degree, called 
Conies (Art. 152), is 

or F{x, y; x,y)=^0. 

Among many ways of treating conies that seems most natu- 
ral which proves itself the best in the study of Quadrics (sur- 
faces of second degree) , namely, to develop the relations of 
the locus to the RL. Let the student recall that 

The Eq. of the polar as to F{x, y ; x,y) = of the pole 
(^1, 2/i) is 

F{x^, 2/i ',^^y) = 0, or F(x, y ; x^, 2/1) = (Art. 69) , 

or {^x^-^7iy^ + g)x-Jr{hx^+jyi-j-f)y-\-gX:,+fyi-^c = 0. (J) 

If the pole be on the curve, the polar is a tangent at the pole. 
The Eq. of the pair of tangents to F{x, y ; x, y) = 
through (Xi, 2/1) is 

F(x,, 2/1 ; a^i, 2/1) • ^'{x, y ; x, y) = \F{x^, y^ ; x, y) ]" (Art. 72). 

By passing to |1 axes through a new origin, 0'{x',y'), Z:, /i, / 
are not changed, but g, /, c are changed into 

g<=-kx'-{-hy'+g, 

f'=hx'+jy'-^f, 

c' = F(x', ?/' ; x', y') (Art. 51). 



If A 



k 


h 


9 


h 


j 


f 


9 


f 


c 



0, the conic breaks up into two RLs. 



If, also, (7= A;/ — Zi^ = 0, the RLs. are 



114 



CO-ORDINATE GEOMETRY, 



91. If {x\ y') and (x, y) be a fixed and a variable point on 
a RL. sloped to the X-axis, the J_ to which is sloped a resp. /5 
to the X- resp. Y-axis, and 8 be the distance between the 
points, then either of these two equivalent sets of relations 
states the Law of Sines of the A PDF' : 



X — x' __ y — y' __ 8 



Put q' = 



sin 0) — 
sin $ 



sin^ 



cos a 



sni (o 



cos (3 



x[ _y — y' _ S 
cos a sin (D 



q = 



sin 0) — 6 _ cos /3 . 



sm 0) sni CO sm w 

sin ^ _ q' 
sin 0) — 9 
Then x = x' + qS, y = y' -{- g'8. 



sinw 



whence s = 




To find the distances from (x', y') at which the RL. meets 
the conic, substitute in F(x, y ; x, y) =0, as in Art. 51 ; 
hence 

{kq'- + 2hqq'-\-jq''-)h' + 2\ (^V-^ %'+ 9)Q 

^(hx'-\-jy'+f)q'lS-^F{x',y'',x',y') = 0. 

A geometric interpretation of this Eq., according to Art. 60, 
lays bare the form and general properties of the conic. 

92. The roots of this quadratic, 8i, 83, are counter (equal and 
unlike-signed) , i.e., (x'^ y') is the mid-point of a chord of the 
conic, directed by 0, when, and only when, 



DIAMETERS AND CENTRE. 115 

{Tix'+ hy'+g)q + (hx' -}- jy' -^ f) q' = 0, 
or (Jcx'-hhy'-i- g) + s{Jix'+jy'-^f) = 0. (K) 

For constant, s, g, and q' are constant, and the Cds. {x', y') 
of the mid-point of any chord directed by 6 are connected by an 
Eq. of first degree; i.e., the mid-points of all \\ cJiords of a 
conic lie on a RL. Such a RL. is named a Diameter. 

By changing $ we change the direction of the H chords, 
change s, and change the diameter ; s, tlien, is the parameter 
of the system of diameters, and since it enters (/r) Hnearly, all 
diameters jjass through a pointy called the centre of the conic. 

This centre is the intersection of the two diameters 

kx' -\- hy' -\-g = and hy' -\-jy' -|-/= ; 

i.e., the point (G : C, F: C). It is in finity or in cc according 

as (7^ or (7=0. Hence conies are named conveniently, 

but not quite correctly, centric and non-centric, according as 
the centre lies in finity or not in finity. 

93. We have got the notion of centre from that of diameter, 
but we ma}' get this from that, thus : 

The coefficients of x and y in Eq. (J) of the polar of (o^i, yi) 
vanish when kxi -\- hyi -\- g = and hxi-{-jyi-\-f= 0^ i.e., 
when the pole is the point {G : C^ F : C) ; but when the coeflfi- 
cients vanish, the intercepts on the axes are both oo, i.e., the 
ML. lies wholly in oc.* Hence (G : C, F: C) is the pole of the 
polar at go. Hence the polar of every point at co goes through 
{G : C, F: C). Now a pole and an}' point on its polar are a 
pair of points to which the section-points with the referee of a 
RL. through the pole and point on the polar are an harmonic 
pair; and since one point of the first pair is at oo, the other 
halves the tract between the second pair (Art. 41) ; i.e., the 
point {G: C, F:C) halves every chord of the referee (conic) 
through it. Such a point is named centre. 

* See Note, page 19G. 



116 CO-ORDINATE GEOMETRY. 

Now take any point on a RL. or polar through this centre ; 
by Art. 71 it is the fourth harmonic to the pole at co, and hence 
halves the tract between the other pair, namely, the section- 
points with the referee (conic) of the E.L. through it and the 
pole. But all such RLs. are 1|, since, as the point glides along 
the polar, they turn about the same point at oo, the pole of that 
polar ; hence, too, they are all conjugate to that polar and no 
other RLs. are ; hence, a RL. through the centre of a conic 
halves a system of \\ chords tuhich are conjugate to it as a polar. 
Such a RL. is called a Diameter conjugate to the chords it 
halves. 

Among all the chords conjugate to any diameter D there is 
one and only one through the centre, which is accordingly^ D's 
conjugate diameter D'. The chords which D' halves are II to 
D. For D passes through the pole of D' at oo , and hence all 
ll's to D pass through that pole at go ; hence all chords H to 
D are conjugate to D', and hence halved hjD'. Hence, conju- 
gate diameters., D and D', of a conic halve each all chords \\ to 
the other. This, indeed, is clear from the fact that the conju- 
gate relation is mutual (Art. 69). 

94. From {K) we see that the direction-coefScient s' of the 
diameter halving chords whose direction-coefficient is s is 

, h -\-hs 1 k-\-hs' 
s'= ; whence s = — - • 

h -\-js h + Js' 

If C=hj-h' = 0, 

then .'=-^, ,^_VRV^ + VJ-^0. 

Vj vj (Vk-hVj's') 

Hence, s' is constant, i.e., diameters of a non-centric conic 
are \\. We may not cancel VA: + \/j'S' in the numerator and 
denominator of s, since it is 0, and to divide b}' has no sense ; 
but s takes the undetermined form ^. To find what this means, 
put s' or its value for s in (K) ; so we get the Eq. of the diam- 
eter conjugate to the chords directed by s, i.e., the II diame- 



CONJUGATE DIAMETEKS AS CO-ORDINATE AXES. 117 

ters. Redncing, we get Ox -}-0y -i-g'Vj -{-fVk = ; but this, 
by Art. 92, is the Eq. of a RL. at oo. Now this RL. at oo goes 
through the centre at oo, and so halves all chords through that 
centre, i.e., halves all the |1 diameters, i.e., is conjugate to 
them all ; and this it does whatever its direction may be. 

Hence, the common conjugate to all diameters of a non- 
centric conic is the RL. at oo, and may be thought H to them 
all. 

95. For a = 90°, q' is 0, and the diameter is 

Jcx' + Jiy'-\-g = 0; 

this, then, is the diameter halving chords H to the X-axis. 
So, too, hx' -^j'y' -\-f= is the diameter of chords II to 
the Y-axis. These diameters are themselves 1| to the Y- resp. 
X-axis when and only when 7i = ; but then they are con- 
jugate, as each halves chords H to the other ; hence, the condi- 
ti07i necessary and sjtfficient that the axes be \\ to a pair of conju- 
gate diameters is /i = 0, i.e., the term in ocy must vanish. 

If the centre be taken as origin (which can be done always 
and only in centric conies) , the new coefficients of x and ?/, 
kx^ + %i + g and hx-^ -\~jyi +/? vanish, and the central Eq. 
becomes 

ka^ + 2 hxy -\-jy^ + c' = 0. 

If, besides, a pair of conjugate diameters be taken as axes, the 
term in xy vanishes, and the Eq. takes the form 

This Eq. is a pure quadratic both in x and in y : to any value 
of either correspond two counter values of the other, each 
axis halving all chords I| to the other. 

96. In general, diameters are oblique to their conjugate 
chords ; are the}^ ever X ? Choose rectang. axes ; then s and 
s' become tan and tan 0', and 

4. Ai k -\-h tan 
tan^'= 

h -f j tan 



118 CO-ORDINATE GEOMETRY. 

When chords and diameter are J_, tan 6- tan ^' = — 1 ; or, on 
reduction 



/i tan 6*^ + /c — J tan ^ — 7i = 0. 

This quadratic in tan has two roots : tan ^i, tan 62^ both always 
real. They yield each an 00 of values of 6*, but as the}^ differ 
among themselves only by some multiple of tt, the two oc's 
determine but two different directions of chords J_ to their 
diameters; also, since tan ^^tan ^2= — I5 these directions 
are J_ to each other : hence, if either be thought as the direc- 
tion of the chords, the other will be the direction of their 
diameter. Again, b}' Art. 52, these two _L directions are the 
ones that halve the angles between the directions fixed by the 
pair of RLs. kx^ -\- 2 lixy -\-jy^z= 0. Hence, there is one and 
only one ])(^i'>'' ^f J- conjugate diameters : the pair halving the 
angles between the Asymptotes (see Art. 97). They are named 
Axes of the conic. 

N.B. Of course, in the non-centric conic only one diameter 
_L to its chords is in finity ; it is called the Axis of the conic. 

97. If the coefficient of the second power of S be 0, one root, 
§1, of the Eq. is go ; i.e., one distance from (a?', ?/') to the conic 
in a direction fixed by 

kq" + 2 hqq' -\-jq'' = 0, or k + 2 lis +J8^ = 

is 00. This Eq. is quadratic in s ; hence there are two such 
directions, which are real and separate, real and coincident, or 
imaginary according as h^ — kj{ov — (7) is > 0, =0, or 
< 0. The conic is named accordingly Hyperbola (excess), 
Parabola (likeness) , or Ellipse (lack) . Denote them by //, Z', 
£. H has points at 00 in two directions, P in o?ie, £ in none 
(real) . 

The co-factor C is called the criterion of the conic. The 
direction in which lies the point at go in P is fixed by the value 
of s : s = V^' : Vj, since kj — /r = ; but this is the 
direction-coefficient of the H diameters ; hence all diameters of 



ASYMPTOTIC DIRECTIONS. 119 

the non-centric P meet it at go, and hence can meet in only one 
point in finity. 

98. If the coefficients of both powers of 8 vanish, then both 
roots, 8i, §2? ^^6 00 ; the coefficient of 8 vanishes when and 
only when both 

lix' -j-hy' -\-g = and hx' -\~jy' +/= ; 

i.e., only when the origin is at the centre ; i.e., in // and £, not 
in P. Hence, two RLs. drawn through the centre in directions 
fixed by k-{-21is -\-js^ =0 meet the centric conic each in 
two points at oo . Now the points at go on a RL. are consecu- 
tive ; hence, these RLs. meet the centric conic at two consecu- 
tive points at GO ; i.e., they touch it at go. 

These RLs. through the centre tangent to the centric conic 
at 00 are named Asymptotes ; they are real in //, imaginary in 
£. Their directions may be named asymptotic. All RLs. 
drawn in asymptotic directions, except the asymptotes, meet 
the conic in one finite, and one infinite, point ; both points are 
real in H^ imaginary in E. 

In the non-centric conic, P, the one diametral direction rep- 
resents two coincident asymptotic directions ; all diameters of P 
meet the curve in one finite, and one infinite, point. 

99. If -2/, E\ C be ends and centre of a diameter, P any pole 
on it, P' the section of the diameter with P's polar, then, by 
Art. 71, J5/, P, ^', P' form an harmonic range ; 

. EF'E'P' ^_^ ^^ PP:P^'=PP':-J5P'. 
FE'-F'E 

C E 



E' F P' 

On compounding and dividing, results 

CP:CE=CE:CP'', ♦ 

i.e., The geometric mean of the central distances of any pole and 
its polar, measured on any diameter, is half that diameter. 



120 CO-OHDINATE GEOMETRY. 

In P the centre C and one end of the diameter, say J5", retire 
to 00 ; hence E' halves PP' outerl}^, and hence E halves PP' 
innerly. 

By definition the poles of a system of |1 chords lie on the 
diameter, D, conjugate to the chords, halving them ; hence the 
tangents at the ends of any one of these chords meet on the 
diameter^ D, in the pole of that chord; if the chord be a diame- 
ter, 7)', its pole is the point at oo on D, and the tangents 
through its ends are accordingi}' I| to each other and to D ; 
i.e., tangents at the ends of a diameter are \\ to its conjugate. 

In P this conjugate is at oo, but the tangent is still || to the 
diameter's conjugate chords. 

100. We have reduced (Art. 94) the Eq. of centric conies, 
H and £ ; to reduce that of the non-centric, /*, take as X-axis 
any diameter, as I"-axis the tangent through its end. Then 
the absolute vanishes, the origin being on the curve ; the 
Eq. becomes a pure quadratic in ?/, since to any value of x 
must correspond two counter values of ?/, the chords || to 
the y-axis being halved by the X-axis ; the term in ic- vanishes, 
since, for any value of ?/, one of the ic-roots of the Eq. must be 
00, one finite, the H's to the X-axis, i.e., the diameters, 
meeting the curve in one point in finity, one in oc ; there remain 
onl}' the terms in x and ?/^, which may be written conveniently 
thus: ?/- = 4g'a5, the Eq. of the P referred to a diameter 
and the tangent at its end. 

The figures on page 121 illustrate fully the foregoing articles. 

101. Let us resume the stud}" of the quadratic in 8. The 
coefficient of 8^ contains A;, /^, J, 6^ and may be written 

^(k, h,j) 6) ; 

then the preduct of the roots Si, 89, i.e., of the distances of 
(x', y') from the conic in the direction 0', is 

F(x',y'',x',y'):cp{k,hj',0') 



FIGUEES. 



121 




122 



CO-OHDINATE GEOIVIETKY. 



The product of the distances of (x", y") in the same direc- 
tion is 

F(x'\y";x",y"):i>{k,JiJ;0'). 

The ratio of these products 

F{x',y';x',y'):F{x",y"-x",y") 

is independent of 0'^ i.e., the same for all directions; i.e., 
the ratio of the iwoducts of the distances of any two points from a 
conic is constant for cdl \\ directions of the distances. 

By taking two fixed directions 0', 0" and one arbitrary point 
(x'',?/'), instead of two fixed points (o^',?/'), {x",y") and one 
arbitrary direction (9', we get as ratio of the product of the 
distances 

cp{h,h,j;0'):H7cJiJ;0"), 

— a result independent of the point (x', y') , the same for all 
points ; i.e., the ratio of the product of the distances of a point 
from a conic, measured in two fixed directions^ is constant for 
all points. 

The interest of these theorems lies mainly in the special 
cases .: 

(1) Take the centre as the point; then the two distances in 
any direction are equal, being halves of a diameter ; hence the 
ratio of the products of the distances from any point to a conic is 
the ratio of the squared diameters \\ to the distances. 




(2) Take the directions tangent to the conic ; then the dis- 
tances are again equal ; taking the second root of the ratio of 



RATIO OF DISTANCE-PRODUCTS. 



123 



the products, we see that the ratio of tico tangent-lengths from a 
point to a conic equals the ratio of the \\ diameters. 

(3) Take as directions those of the diameter through the point 
and its conjugate chord ; then the distances on the chord are 
equal, and those on the diameter are its segments ; hence the 
square of any chord varies as the product of the segments into 
ivMch it cuts its conjugate diameter. See the figures. 





102. B}' Art. 51 A:, 7i, j are not changed by a change of 
origin ; to find how they are changed by a change of axial 
directions, we might use the general formulae of transformation 
(Art. 21) ; much neater, however, is this method of Boole : 

The transformation formulae being homogeneous in Cds., in 
passing from axes X, I^to axes X', y, inclined w resp. o>', the 
expression kocr -\- 2 hxy -\-jy' changes into 

so that A:aj2 + 2 hxy -\-jy^ = k'x"- + 2 h'x'y' -hfy'^. 

Also XT -{- 2xy cos o) -{- y^ = x'^ -{- 2 x'y' cos (o' + y'^, 

since each is the squared distance of the same point (x, y) 
(ic', y') from the common origin. Add this Eq., multiplied by 
an arbitrary /x, to the first ; there results 

(Jc + fjL)x^ -f 2 (7i + /x cos w)xy + ( j + fx)y^' 

= {k'+ ^)x'' + 2 (/.'+ /. cos coO^-^y + (/+ /x)2/'^ 

Each side of this Eq., equated to 0, represents the same locus : 
a pair of RLs. through the origin (Art. 44) ; if /x be chosen so 



124 CO-ORDINATE GEOMETRY. 

that the RLs. fall together, each side becomes a perfect square ; 
i.e., the same values of fx make both sides perfect squares ; i.e., 
the roots, /xj, /xo, of the two Eqs. : 

(k + /x) (j + /x) = (7i + /x cos <o)2 
and (A;' + />t)(/ + /x) = (7z'+/xcoso)')^ 

are the same; i.e., corresponding ratios of the coefficients of 
the powers of fx in the two Eqs. are equal ; i.e.. 



(J*^ +i — 2/i cos o>) : sinco"= {k' -}- f — 2 h' coso)') : sin to' , 



(kj — h~) : sin w" = (k'j' — h'-) : sin to'" ; 



i.e., the ratios (k -i-j — 2hcosw) : sinw and (^J — 7i^) : sin oo 
are unchanged by any change of axes. 



Geometric Interpretation. 

Suppose the central Eq. of a centric conic brought to the 
form : 

kx^ + 2 hxy -\-jy^ = 1 ; 

then are — -, — — the intercepts on the axes, and are half-diam- 

Vk Vj 
eters. 

1. For (0=90, from the above, k-\-j is constant ; i.e., 
the sum of the squared reciprocals of two rectang. diameters is 
co7istant. 

2. For conjugate diameters taken as axes, h = ; hence 

'^ and - ^ are constant ; hence their quotient - + -7 •> or 

sin 0) sin o) ^ J 

the sum oftivo squared conjugate diameters, is constant. 

Also, by inverting and taking the second root, — — is con- 

stant; i.e., the area of the parallelogram of two conjugate half- 
diameters is constant. 



THE FOUPv CENTKICS. 125 



CHAPTER y. 

SPECIAL PROPERTIES OP CONICS. 

Centric Conies : Ellipse and Hyperbola. 

103. The Eq. of the centric conic referred to conjugate 
diameters is 

A:'aj2 -]- jy -\-c' = (Art. 94) . 

Two general cases present themselves : 

I. k' and j' like-signed, say both + ; then the criterion 
C ^kj — 7i^ = k'f — > ; hence tJie curve is an ellipse. Under 
this head are two special cases : 

(1) c' < ; then the ellipse is real, denote it by E. 

(2) c' > ; then the ellipse is imaginary, denote it by £'. 
For clearly no real values of x and y satisfy its Eq. 

II. k' and j' unlike-signed, say k' + and f — ; then the cri- 
terion C^kj — h^ = k'f — < ; hence the curve is an 
hyperbola. Under this head are two special cases : 

(1) c'<0 ; then the hyperbola \^ primary, denote it by H. 

(2) c' > ; then the hyperbola is secondary, denote it b}- H'. 

11 k' j' 

Now write — , — for , — — ; on observing signs there 

a'" 6'- c' c' 

result these Eqs. of E, E', H, H' , referred to conjugate diameters: 

9 9 9 

9 9 9 9 

^_2/"'_i or y- _ -. 



126 CO-ORDINATE GEOMETRY. 

To denote that the pair of rectang. conjugate-diameters, or 
the axes of the conic, are taken as Cd. axes, clroio the primes 
from a and b. 

104. Thus far little reference has been made to figures, for 
the shapes of the curves were supposed unknown, and it was 
deemed important to illustrate how the properties of curves 
may be deduced while their forms are A'et unknown. Eeason 
far outruns imagination. We may reason correctly about 
forms we cannot imagine at all. But we may now find out the 
shapes and draw the figures of three of the above curves. 
Only the imaginary £' is unrepresentable in our plane. 

Putting y=0 in the Eqs. of £, //, //', there results 

x= ± a, x= ± a, x—±ia; 

i.e., all three cut the X-axis on each side a from the origin 
(centre) : £ and M in real points, //' in imaginary points. 

So, if X = 0, then ^ y = ±b, y=±ib, y = ±b ; 
i.e., all three cut the I"-axis on each side b from the origin 
(centre) : £ and f/' in real points, // in imaginary points. 

It is common to assume a > 6 in the £ ; then 2 a and 2 b are 
called axes 7najor and minor of the £. 2 a resp. 2 6 is the real 
(commonly called transverse) axis of H resp. H' ; 2 ib resp. 2 ia. 
is the imaginary axis of H resp. //'. The real axis 2 6 of M' is 
often called, though hardly properly, the conjugate axis of H. 

Plainly, like results hold when a and b are primed ; i.e., when 
any pair of conjugate diameters are taken as Cd. axes ; £ cuts 
both in real points like-distant from the centre ; H cuts only 
one of two conjugate diameters in real points, while H' cuts the 
other. Hence, while all the real ends of one system of diame- 
ters lie on //, all the real ends of their conjugates lie on ff' ; and 
conversely. Hence H' is commonly called the conjugate of H ; 
strictly each is the conjugate of the other. 

Since the rectang. Eqs. are pure quadratics in both x and ?,', 
each curve is symmetric as to each of its axes. 



6^ = 


cC^-V" 


■ a? ' 


0- — 


cr + &' . 




9 ' 


rP- — 


«' + y . 



THE CENTRICS TRACED. 127 

Clearly a and 5 are the greatest values of x and ^Z in £ ; a is 
the least value of x'vuH^h the least value of y in. H' . 

105. If we pass to polar cds., putting p cos (9, p sin (9 for x^ y^ 
there results on reduction and inversion : 

for £, p^ = ^ , where 

1 — e^ cos ^ 

— h^ 
for ^, p- = ^ , where 

1 — e^ cos e?" 

for /^', /o" = ^^^:3- , where 

1 - e2 cos r ^^ 

as central polar Eqs. of £, /^, /^', one side of the real axis of H 
being polar axis. That of H resp. //' is got from that of £ by 
simply changing the sign of 6^ resp. a-. The geometric mean- 
ing of e^, used here for shortness, will be seen later. 

These Eqs. are pure quadratics both in p and in cos^, also 

2 2 '^ 

cobO =qos{ — 0)^ = cos^tt — Oy, and two counter p's make a 
diameter ; therefore, 

Diameters like-sloped to an axis are equal, and equal diameters 
are like-sloped to an axis. 

106. Let us trace £. For ^ = 0, p = a; as ^ increases 
to -, p decreases to 5 ; as ^ increases to tt, p increases to a;. 

as increases to — , p decreases to 5 ; as 6 increases to 27r, p 

increases to a. The greatest resp. least diameter is 2 a resp. 

26. 

\ I 

In H, for ^ = 0, p = a -, as ^ increases to cos"^ -, p 

e 

increases to go, all values of p in H' being meanwhile imagi- 
nary ; as $ increases from cos"*-^- to - and thence to tt — cos'^ -, 

e 2 e 

p in ff' decreases from oo to 6, and thence increases to co, all 
values of p in // being meanwhile imaginary ; as increases 



128 CO-ORDINATE GEOMETEY. 

from TT— cos^^- to tt and thence to 7r + cos~^-, p in H 

e e 

decreases from oo to a, and thence increases to oo, all values of 
p in //' being meantime imaginary ; as 6 increases from 

1 3 TT 1 

IT + cos~^- to -— and thence to 2 tt — cos~^-, p in //' decreases 
e 2 e 

from GO to &, and thence increases to oo, all values of p in H 

being meantime imaginary ; as increases from 2 tt — cos^^ i 

e 
to 2 TT, p in // decreases from co to a, all values of p in //' beino- 

meantime imaginar}-. 

Y 




cos ^- 

with their counters corresDond to 



The two directions 



and 



= 7r — cos - 



MAJOE AND MINOR CIRCLES. 



129 



tan ^ = 4- - •» and tan 0= , 

a a 

which are the direction-coefflcients of the asymptotes : 

,2 






0. 



Hence the two H^s have common asymptotes, and along these 
asymptotes they close in upon each other at go. 



107. Solved as to y resp. x the Eq. of £ is 



o /— 2 9 

ye = - ^ a —x-" resp. 
a 



^e = 7 V6^ — 2/^. 



Now y^ = -\/o? — x^ resp. Xc= V6^— 2/^ is the Eq. of 
a circle about the centre (origin) , radius a resp. 6, which may 
be called the major resp. 7)iinor circle of the £. For any value 
of X the corresponding values of y in the E and the major circle 
are in the ratio 2/e = 2^e = ^ • o!" Hence the E is the ortJiogo- 

nal projection of its major circle under the X cos""^-* 

a 





_Z 








p' . 


^ 




Pf^ 




/ 1 ' 






VA^ 


^:^ 








<p^ 


'- 














y 


^. 



So the minor circle is a like projection of the £. Think the 
surfaces of £ and the major circle made up of elementary trap- 
ezoids, or covered with threads ± to the common diameter, cor- 



130 



CO-OEDINATE GEOMETRY. 



responding elements of the two surfaces will have the fixed 
ratio h :a\ hence the whole areas will have that ratio ; i.e., 



area of £ = - • ira^ = vab = V-Tra- • 7r6^ 
a 

equals the geometric mean of the areas of major and minor circles. 

The student can easily convince himself that the ratio of the 
projection of an}' plane area to the area projected is the cosine 
of the angle of projection. 

b 



The Eqs. of £ and H solved as to ?/ : 



o /-r> : 

y = -^cr — X- 
^ a 



and 



y = - 'Vx- — cr declare that an}^ ordinate is the -th part of 
a a 

the geometric mean of the segments into which it cuts the major 
axis; for the segments are a-{-x, a — x in the £, and x-{-a, 
x — a in the //. In the £ the section is inner; in the H it is 
outer. 

For a^=b, the £ reduces to a circle or equiaxial £. The 
equiaxial //is x^ — y^= cr. It corresponds to the circle, and is 
called also equilateral or rectangular, since its asymptotes are 
_L. It is congruent with its A^' : xr — y^=—a^, and falls 
on it when turned through 90°. As in the circle, so in the 
equiaxial M, any two conjugate diameters are equal. 




108. B}' Art. 93 the direction-coefficients of two conjugate 

diameters are connected bv the relation, s' = ; or, 

h -i-js 



CONJUGATE DIAIVIETEES. 131 

if conjugate diameters be axes, and so A = 0, ss' = ; 

J 
i.e., their product equals the negative ratio of the coefficients of 

oc^ and y'^ ; hence, in the present form of the Eqs. of E resp. H^ 



.1 
a'^ ~ a 



b" 



ss' = ——^ resp. ss' = — , 



7 2 7 2 

or tan ^- tan ^'= resp. =H — -. 

a"* cr 



Hence tan 6 and tan 0' arc unlike-signed in £, Hke-signed in N ; 
i.e., & and 0' lie in adjacent quadrants in £, in the same quad- 
rant in // ; i.e., of two conjugate diameters of an £, one lies 
in first and third, one in second and fourth, quadrants ; but of 
an ^, both lie in first and third or both in second and fourth. 

In £, if tan^= ±-, tan^'=:if:-: i.e., when one of 

two conjugate diameters of an £ is one diagonal of the rectangle 
of the tangents H to the axes, the other is the other diagonal. 
This pair of diameters of the £ are named equi-conjugate. 

In H, if tan^=±-, tan^' = dz - 5 i-e., tivo conjugate 

(Jj Lb 

diameters fcdl together on each diagonal of the rectangle of the 
tangent I| to the axis; i.e., on each of the asymptotes; hence 
each cLsymptote is a self-conjugate diameter. 

In £, as 6 increases, 6^ increases ; in /^, as 6 increases, 6' 
decreases. 

109. The Eq. of the tangent to £ or H is (the upper sign 
going with £) 

_j_ ^ jjj^ = 1, or -!— ± --^-^^ = 1 . 
a~ Q- cr 0' 

X Xi 

The Eq. of the diameter through (x, y) is - = — . Its con- 

y Vi 

jugate is 1| to the tangent, and goes through the centre ; hence 
its Eq. is 

9 79 

cr Q-' 



132 CO-ORDINATE GEOMETRY. 

To find the Ccls. Xo, y^ of an end of this conjugate, combine 
its Eq. with the Eq. of the curve, thus : 



+ ^_ = 1 or — ^ J ^1- + :rL L = 1 
x,'b' b' ' b-'x^b' a'i 

For £ resp. // the parenthesis J J is +1 resp. —1 ; hence, 
for £", 

_ a L ^ . 

^2 — -F tVii y^ — ± ~^i 5 
6 a 

for //., 0^2 = ± ^V2/i^ 2/2= =F^'- ^1- 

Again we see onlj' one of two conjugate diameters has real ends 
on an H. 



110. Plainly, if the x (or y) of one end of a diameter is 
known, the diameter itself is known as one of two equal diame- 
ters like-sloped to the X-axis ; hence we can express any 
squared half-diameter through the x of its end. If (o^i, y^) be 
the end of a' in an £, then 

By Art. 102 a'^ + 6'- = a^ + &2, hence 6'- = a^ - e^a;^. 
Now b^^ = X2+y2\ hence, by Art. 109, 

2 7 2 

a o , t) 9 9 9 

T-9 2/i + -9 ^i" = a' - e'iK'. 
b'^ a^ 

Changing the signs of 5^ and 6'^, we get for the /^, 



a'- = —6^ + e-iCi 



2^ 2 
5 



r^ = ^2 _ ,2^^2 _ -("-l!.^,^ + S""'')' 



111. By Art. 102 the area of the parallelogram of two conju- 
gate half -diameters, and therefore its fourfold : the area of the 
parallelogram of the tangents through the ends of two conju- 



CONJUGATE DIAMETERS. 



183 



gate diameters, is constant. If <^ be one ^ between these diam- 
eters, then this area is 4 a'b' sin cf) and = 4 a6 ; whence 



sin di = 

^ ,ihi 



ah 

cJb' 



Hence </) is least when a'b' is greatest ; and a'6' is greatest when 
a2 -f- 52 _,_ 2a'6', which = a'^ -\-¥'-2 a'b', which = (a' - b'f, 




is least; i.e., when a' = b' ; i.e., when i^/^e diameters aretJie 
equi-GonJugates. For equi-conjugates, 

a6 2 ab 
Vb' 



sin ^ 



a2 + 6^ 

Of course this last reasoning is necessary and applicable only in 
case of the £; in the H the ^ be- 
tween conjugates is least when they 
fall together in an asymptote. 

If 2 a', 2 b' be two conjugate diam- 
eters, 2_p the distance between the 
tangents |1 to 2 a', then the constant 
area is 

2a' .2p — 4.ab, or 

i.e., the distance from the centre to a tangent is a fourth proctor- 
t toned to the half-diameter \\ to the tangent and the half -axes. 




134 



CO-OHDINATE GEOMETRY. 



112. The definition of the normal (Art. 73) yields as its Eq. 



a 



a- 



_ (i» — xOt- (2/-2/i) = 0; 01% -.xJ^ -'y = oj'^:W. 



X 



2/1 



it/l 



2/1 



The intercepts of the tangent on the axes are — and ± — ; 
those of the normal are ^ 



2/1 



«' =F &' A 

— — — 0^1 and 






The product of corresponding intercepts of T and -^ is a 
constant : ± (ctr ^ Ir) . 




The intercepts on the tangent between the point of touch and 
the X- resp. Y-axis may be named X- resp. Y-tangent (lengths). 

To like intercepts on the normal like names are given. 

The projections of these tangent and normal-lengths, each on 
its own axis, are named sub-tangents and sub-normals. 

The following table needs no explanation : 



X-subtangent 



a. 



CCi — • 



a? — x^ _ a^yi 



*j(j-\ 



±h'~XT^ 



y-subtangent = — — y-^ = 

2/1 2/1 



_ ^^j-J/i _ ± b^^i 



«'2/i 



2/i; 



Xi ; 



(1) 



(2) 



TANGENT AND NOHMAL. 



135 




a^ zr. ffi 7)2 

X-subnormal = x-^ ± — Xi = ±-—Xi; 



a" 



2 T- 7.2 



a- 



F-subnormal =y,- ^-'^^^Vi = ±~yi\ 






X-tangent = { ., + g^:,, jLg {|^. + |,, 



= ^1.6'; 
hxi 



y-tangent 






^i^^=7^Mf^2// + -^^: 



«2/i ( y^ 



a" 



—1.6'; 
«2/i 



X-normal 



2/,^ + ^a.,4^=M«\,2 



or ) a I Ir' a^ j 






r-uo™al ={.,^ + |;,.^}'=^{|.,. + |,,.}» 



= ^■6'. 



(3) 
(4) 



(5) 



(6) 



(7) 



(8) 



136 CO-OBDINATE GEOMETKY. 

Hence these results are evident : 

(1) Product of ST's = product of SJST's = product of Cds. 
of point of touch. 

(2) Product of T's = product of N's = squared half -diam- 
eter II to tangent. 

(3) Product of X- resp. Y-normal by central distance of tan- 
gent = 5^ resp. a^ 

113. As the tangent is but a special case of the polar, so 
the normal may be subsumed under the more general concept 
of a _L through the pole to the polar. In lieu of a better, give 
this _L the name Perpolar. Since the Eq. of the polar has 
the same form as that of the tangent, the Eq. of the per- 
polar has the same form as that of the normal. As the pole 
glides along a EL., the polar turns about (envelopes, en- 
wraps) a point ; but in the Eq. of the perpolar, which may be 
written 

a?yiX qi Irx^ = (a^ q: b^)xjyi, 

the parameters Xi, yi do not appear linearly ; hence, when con- 
nected by some linear relation, it will not in general be possible 
to eliminate one from the Eq. of the perpolar and leave the 
other in first degree only; i.e., as the pole ix^.y-^) glides along 
a RL., the perpolar will not in general turn about a point, 
but about some curve. But in three cases it is possible : when 
Xy is constant, when y^ is constant, when ?/i : x^ is constant ; 
i.e., when the pole moves on a RL. \\ to either axis or through 
the centre^ the perpolar turns about a pointy the perpole of the 
RL. 

If the RL. be H to the Y'-axis, x^ is constant, and the pole of 
the RL is on the X-axis, distant a^ : x^ from the centre ; then 
the perpole is also on the X-axis, distant e'^x^ from the centre. 
(This is seen at once on writing the Eq. of the perpolar 
thus : 



FOCI OF THE CURVE. 137 



T b^Xiif + yi(a^x — a^ ^: b'-x^. For x^ constant, this is the Eq. 

of a pencil of RLs. whose base-lines are y = 0, i.e., the 

ct m b 
X-axis, and a-x — (a^ ip 6-) Xj = 0, i.e. , x = — ^^ — x^ = e^Xy^ ; 

Cv 

. • . the perpole is [e^x^^ 0] ) . The product of these two central 
distances is the constant aV ; hence, poles and perpoles of ULs. 
II to the Y-axis form an involution of points on the X-axis, 
whose centre is the centre of the conic, whose foci are distant ae 
from the centre. 

Likewise it is proved that poles and perpoles of RLs. [] to 
the X-axis are in involution on the T-axis, but the constant 
product of distances of a pair from the centre is — cre^. Hence 
if either pair of foci are real, the other are imaginary. 

The student will readily see that poles and perpoles of all 
RLs. through the centre lie on the RL. at oo. 

114. The foci of the involutions on the axes are called foci 
of the curve; hence a centric conic has four foci: two real, two 
imaginary. They enjoy important properties. 

The central distance, ae, of a real focus is called the linear 
eccentricity of the conic ; e itself is the eccentricity proper. It 
is the ratio of the central distance of a focus to the half-axis on 
which the focus lies. Now in case of the imaginary focus the 
central distance of the focus is imaginary in both £ and H ; but 
the half-axis is real in £ and imaginary in H ; hence their quo- 
tient is imaginary in £but real in ff ; i.e., one eccentricity is real^ 
one imaginary in £, both are real in H. 

The j^olar of a focus is called a Directrix. Suppose the real 
foci (ae, O) , ( — ae, O) on the X-axis ; the directrices are 

a 

3C = ± — 

e 

By Art. 113 a focus is a double point in which have fallen 
together pole and perpole of a certain RL., the directrix. As a 
pole glides along this directrix, both its polar and its perpolar 
turn about the focus cdtcays _L to each other. 



138 



CO-OEDINATE GEOMETRY. 




Call the tract from a focus to a point a focal i^adius of tliat 
point, and any RL. through a focus a focal chord. From any 

point P' of the directrix draw a 
chord cutting the conic at I and J' ; 
to the pole Q of this chord, and 
from the focus P, draw FQ cut- 
ting the chord at P. By Art. 71, 
since the polar of P' is PQ, P', 
J'-, P, I form an harmonic range ; 
hence F\P'I'PI\ is an harmonic 
pencil. But FP' is clearly the per- 
polar of P', as this perpolar must 
go through P' and through F. 
Hence FP and FP' are J_ ; hence 
they halve the ^s of FI and FI' (Art. 41). Hence, the 
focal radius of the pole of a chord haloes the ^ at the focus 
subtended by the chord. 

115. By Art. 113 polar and perpolar cut the axis of involu- 
tion in a pair of conjugate points, harmonic with the foci ; hence 
the focal radii of the intersection of polar and perpolar form 
with these two an harmonic pencil ; and since these two are _L, 
they halve the angles beticeen the focal radii. 

When polar and perpolar are tangent and normal, their inter- 
section is the pole, the point of tangence on the conic ; hence 
tangent and normal halve the ^s ofthefoccd radii of the point of 
touch. 

The normal halves the inner resp. outer !^ in the E resp. H ; 
hence an E and an H with the same foci, i.e., coiifocal, are _L to 
each other. 

116. These relations of position imply several relations of 
size : the intercept between the foci being 2 ae, the X-intercept 
of the normal being e^x (Art. 112), the segments of the focal 
intercept are ae + ^'^ii cte — e-% in £, where the normal cuts it 
innerly, and e^x^-f ae, e~Xi — ae in H, where the normal cuts it 
outei'l}'. 



PEEPOLES AND PEKPOLAES. 139 

Hence, as the focal radii r^ r' are proportional to these 
segments, 

r a + exi _ r ex^ + a . 
— T- = m the c , and —r = m the H ; 

r a — exi r exi — a 

or — — = ■ ni the £, and — ■ — = in the H. 

r — r' 2 eXi r — r 2 a 

But, plainly, 

r^ — r'- = (ae + %)^ — (ae — Xi)^ = A aex^, 

or (r + ^'') (^' "''^O = 2a- 2ecci ; 

hence r -\-r'=2 a, r — r'=2 eXi, r = a-\- ex^, T'=a — e%, 

( 9 9 9 7 (9 

rr'= a- — e-xf = 6'- ; 
resp. r + ^'' = 2 e%, 7' — r' = 2 a, r = ex^ + a, r' = exi — a, 
rr'= e-a?i^ — a" = — &'^. 

Or, m ^Zie £ resp. ^, ^/^e s?^m resp. difference of the focal radii 
of a point is a constant, namely, the major resp-. real axis; and 
in both E and H the product of the focal radii of a point is the 
squared half -diameter conjugate to the diameter through the point. 

The distance of {x-^^ y^) on the £ from the directrix x=^~ 

is clearl}^ - — ic^, and the distance of the same point from the 

focus is a — ex^ ; i.e., in the £ the ratio of the distances of a point 
from focus and directrix is a coyistant^ the eccentricity. Plainly 
the like holds for the H. In £ this ratio is < 1, in /^ it is > 1. 

Let the student show that the locus of a point the sum resp. 
difference of whose distances from two fixed points is constant 
is an £ resp. H ; also, the locus of a point the ratio of whose 
distances from a fixed point and a fixed E,L. is a constant 
< 1 resp. > 1 is an £ resp. H. 

The sum resp. difference of the focal Xs on a tangent is twice 

the central _L on the tangent; i.e., it is 2a' — ; the sum resp. 
difference of the focal radii is 2a; also, the ratio of focal _L 



140 



CO-OEDINATE GEOMETRY. 



to focal radius is the same for the two foci, being the sine of 
the slope of radius to tangent ; hence this ratio, this sine is — • 




Dividing the central ± on the tangent by this ratio, we get 
the central distance to a tangent, measured 1| to a focal radius 
to the point of touch, namely, a. Hence the locus of the section 



ASYI^IPTOTIC PROPERTIES. 



141 



of a tangent and a diameter \\ to a focal radius to the ])oint of 
touch is the major circle. The same major circle is the locus of 
the foot of the focal _L on the tangent; for the As PF'JS^, 



MOF are similar, since 



OF 
031 



= e and 



F'JY 
F'F 



ae 



e-x 



a — exi 



1 



= e. 



Asymptotic Properties. 

117. Thus far the properties of £ and H haye corresponded ; 
but the asymptotic properties of the // have no real correspon- 
dents in the £, as the asymptotes of the £ are imaginary. 
Accordingh', in what follows, reference is to the H alone. 



If ^ ^ — 1 i^o +1-.0 u ^ — yi — 

a'^ 0'^ a 



— -^ = 1 be the //, — — -^ = are its asymptotes. 



X a' 

or - =—; 

b' 



is clearly the 



One of these, -^ = 0, 

ah y 

central diagonal of the parallelogram of the conjugate half- 
diameters a', 6' ; the other is the central || to the other 

diagonal, -^-f ^= 1. 
w o' 

Hence, giyen a j^^cwV of conjugate diameters^ we can find the 
asymptotes; or, giyen the asymptotes^ we can find the conjugate 
to any giyen diameter. 




142 CO-ORDINATE GEOMETRY. 

By Art. 42 the RLs. x = 0, y = 0; y = ^x, y = -^x, 

a' a' 

i.e., the asymptotes and any pair of corijugate diameters form 

an harmonic pencil.* Hence tlie asyrtiptotic intercept of a \\ to 
any diameter is halved by its conjugate diameter^ and the inter- 
cept between two conjugate diameters of a || to one as3'mptote 
is halved by the other. As a special case, the asymptotic inter- 
cept of a tangent is halved at the point of tangence. 

Since the same conjugate diameter tliat halves the intercept 
between the asymptotes also halves the chord of the curve, 
clearly the intercepts between the asymptotes and curve are = , or, 

From the Eqs. of the curve and the asymptotes there follows : 

y^ = 7njS = mS' = — X, 
a' 

1.1 



y^z= mC= mC'= — ^x~ 



a' 



.•.CjS-CS'=0'jS'' C'S = b"; 



i.e. , the product of the distances of any point of an H to the asymp- 
totes^ in any direction, equals the squared \\ half -diameters of the 
H. Clearly C8 can be made large, and so CS^ small, at will. 

118. The tangent-intercept between the asymptotes TV 
being halved at the point of touch A\ the A TOT = 2 TOA'= 
the parallelogram A'OB'T of the conjugate half -diameters, a' 



* Hence conjugate diameters form an Involution of which the asymptotes 
are the double or focal raijs. Like may be easily proved of the conjugate 
diameters and imaginary asymptotes of the £ by noting Art. 107 and form- 
ing the Determinant of Art. 47. 



POLAR EQUATlOi^S. 143 

and 6', i.e., = the constant ab. From A' draw to each asymp- 
tote a II to the other, and call them ic and v ; they are the Cds. 
of A', the asymptotes being axes. With the asymptotes they 
form a parallelogram which is clearly half the A TOT' ; hence, 
if <^ be the ^ of the asymptotes, we have itv sin ^ = ah :2. 
But by Art. Ill, since the asymptotes of //fall on the equi- 
con jugate diameters of £, 

. ^ 2ab cr + h- 
sm</) = — ; .'.uv = ■ 

This, the Eq. of the H referred to its asymptotes, says the 
parallelogram of asymptotic Cds. of a point is of constant 
area. 

Focal Xs on the asymptotes are clearly equal ; the asymptotes 
being tangents, their product is — 6"- (Art. 112) ; hence each is 
b in length, but they are counter-directed. This is also plain at 
once from ti'igonometric considerations. 



Polar Equation of Centric Conic. 

119. Take the right focus as pole, the right X-direetiou as 
polar axis ; then, by Art. 116, p=za — ex, resp. p = ex — a 
in £ resp. M ; x being here reckoned from the centre, 

X =zae-\- p cos ; 

, a ( 1 — e-) b^ :r— 7^ 

hence p = — ^ = — : 1 -{-e cos 9 

1 -f e cos a 



a(e--l) b' -, 2 

resp. p = — ^ '- = — : 1 — e cos 0. 

1 — e cos a 

The expressions for the right focal radius being unlike in £ 
and f/. we might have expected these Eqs. of £ and H to turn 
out unlike. This makes the Eqs. just found unhandy. But the 
one expression for the left focal radius is p = a-\- ex. 



144 CO-ORDINATE GEOMETRY. 

The left X-direction being taken as polar axis and reckoned 
clockwise being taken as positive, we have x = — (ae + pcos 0) ; 

a(l — e-) , b^ - — , -^ . r- ,, 

. . p = — ^ ~ — ± — : 1 -f- e cos m £ resp. H. 

1 + e cos a 

These Eqs. of £ and //, like the central Eqs., differ only in 

the sign of b^. For ^=^or— ^, p= ± — . Hence — is 

2 2 a a 

the focal chord _L to the axis; it is called the parameter or latus 
rectum* 

It is interesting to trace the curve from the polar Eq. 

The corresponding values of p are unlike-signed in £ and // ; 
in E p is measured bounding the ^6*, in // it is measured counter 
as long as p falls out negative. The value of e is not the same 
in the two Eqs. ; in £ it is < 1, hence the divisor 1 +ecos^ 
remains throughout + and finite ; but not so in //. 

For ^ = 0, p = a — ae ; in E this is + , but in // it is — ; 
hence it is reckoned leftward in £, rightward in M. As 

increases to cos~^( J, p traces out (by its end) the left lower 

branch of H. For 6 = cos V ], p = — oo, and is drawn in 

the left loicer asymptotic direction. Just here p changes sign, 
and beginning at + oo, traces out the right upper branch of H 
till ^ = TT, when p sinks to a-\-ae at the right vertex; 
thence stajing +, it rises to + QO? tracing out the lower right 

branch of M till reaches tt + cos"^-. Here again p changes 

sign, becoming — oo, and as increases to 2 7r, it traces out the 
left upper branch of //, reaching the left vertex as it reaches 2 tt. 
The student himself may readil}' follow the course in the £. 

It is noteworthy that the right upp)er branch of H is thus seen 
to be continuous (in go ) with the left loicer, and the right lotver 
with the left upper. We are forced to think the H thus by this 
reasoning also : The asymptote touches the // at oo in two 
counter directions ; hence unless it meets each branch in the 
same two consecutive points at go, it must meet H in four points, 



NON-CENTRIC CONIC. 



145 



which is impossible. While, then, to imagination the H con- 
sists of two distinct branches, to reason it consists of one 
branch closed in two directions (the asymptotic) at co. 




Non-Centric Conic : Parabola. 

120. By referring P to a diameter and the tangent through 
its end as X- and Y-axes, its Eq. is brought to the simplest 
form, 



y^ = 4: q'x 



(Art. 100). 



One and only one diameter is JL to its conjugates (Art. 94) ; 
it is called principal diameter or axis of P. The Eq. of P 
referred to these J_s is called the vertical Eq. of /*, the origin 
being the vertex, and is written y^ = 4:qx; 4 g' is called par- 
ameter of the corresponding diameter, 4g is 2^rincipal parameter, 
or simply parameter. From these Eqs. we may now draw out 
all the properties of the P, as is done in most texts ; but another 
method seems directer. 

According as A^ — Ay is < 0, =0, > 0, the conic is £, P, H . 
Hence, am' two of the S3'mbols k, h,j being held fast, as the 
other changes, the conic becomes in turn an £ of this or that 



146 CO-ORDINATE GEOMETRY. 

shape, a P^ and an H of this or that shape. P is thus seen to 
be a critical curve between £'s and //'s, a border or li7}iit of the 
two. What kind of a limit, we shall see. 

For this investigation of the relation of P to E on one hand 
and // on the other, the central Eq. of £ and H is ill suited, as 
the central Eq. of P is unmanageable, the centre being at oo. 
Since the vertical is the simplest Eq. of P, let us move the 
origin to (say) the left vertex in £ and //. The Eq. becomes 



U/ CI , If -t 9 ni / r\ Q\ 

^±^=1 or y- = —^{2ax-x-). 

a- 0^ a^ 

Here Jc= ±-—, h = 0^ i=l; hence, if either E or f/ 

cr 

7 2 

is to pass over into P, -^ must vanish to make h' — Jcj = ; 

cr 

2 Ir 
and — must stay finite to make the Eq. 2/" = 4 qx. Now 
a 

— or 4(/ is the focal chord _L to the axis ; hence — or 2g is 
a a 

7 2 

the ordinate at the focus, 2//, and — or q is the focal abscissa £cy, 

^ Cv 

or distance of the focus from the vertex. Accordingly, we keep 
2 52 yi 

finite by holding the focus and vertex fixed ; to make — - 

a cr 

vanish, we must let a increase toward 00 ; i.e., let the centre^ 
and with it the other focus, retire to 00. But then 

a^ q: 6^ . 52 

\ =lT — = 1, or e^=l. 
a-' a^ 

Hence we may treat P as an £ (or H) whose pa,rameter has 
kept constant tuhile its centre and one focus have retired to 00, or 
as an E (or H) whose eccentricity has increased (or decreased) to 
1 . The properties of P are the properties of £ (or H) at this 
limit, viz. : 

121. P is symmetric as to its axis (Art. 104). 

P is the locus of a point equidistant from focus and directrix. 



P AS THE LOnT OF £ AXD H. 



14' 



The vertex (origin) is distant q from the directrix ; hence ciny 
point (xi, 2/1) of P is distant Xi-\-q from focus and from direc- 
trix. 

The poles of all H chords lie on the diameter of those chords 
(Art. 98) . Also a pole P, the intersection M of its polar and the 
diameter through it, and the intersections J, I' of the diameter 
and the curve, form an harmonic range ; and as I' is at go, /is 
midwa}' between P and M'^ i.e., the intercept on a diameter 
between a pole and its polar is halved by the curve P. . 

If the diameter be the axis, I is the vertex, and the inter- 
cept is the subtangent ; i.e., the subtangent is hcdved at the 
vertex. 

Hence the subtangent is 2 x' long ; also it is cut by the focus 
into segments Xi + q and x^— q. Hence the focal distance of the 
intersection of tangent and X-axis (axis of P) = the focal dis- 
tance of the point of tangence = o^ -f- g. 




Hence the focal X on. the tangent halves the tangent-length ; 
so too does the vertical tangent ; hence the locus of the foot of 
the foccd _L on the tangent is the vertical tangent. 

Further, clearly the focus halves the distance between the 
intersections of tangent and normal with the axis ; hence the 



148 CO-OEDINATE GEOMETRY. 

whole intercept is 2 {x^ + q) ; on taking away the subtangent 
2xi there remains the subnormal = 2q; i.e., tJie subnormal in 
P is the constant half -parameter. 

Plainly a circle about the focus and through the point of touch 
goes also through the intersections of tangent and normal with 
the axis. 

Polar and perpolar and focal rays through their intersection 
form an harmonic pencil (Art. 114). The second focus being 
at 00, the second focal ray is the diameter through the intersec- 
tion. Hence any polar and perpolar halve the ^s between the 
focal ray and the diameter through their intersection. As a 
special case, the tangent and normal halve the ^s between the 
focal ray and the diameter through any]point of a P. Hence, too, 
since the diameter meets the axis at oo, the focus halves the 
axial intercept betiveen the conjugates: polar and perpolar, or 
specially, tangent and normal. This has already been proved 
geometrically in the special case. 

122. All these relations are readily drawn out from the Eqs. 
of tangent and normal : 

2/2/1 = 2 g(a; + x^) and (2/ — 2/i) 2 ^ -f yi(x - x,) = 0, 

a useful exercise left for the student. Eliminating Xi by the 
relation y^ = 4 qx^, we get 

yyi = ^q^ + Y ^^^^ (2/ - 2/1)2 g + yjx - 1^- j = 0.. 

Solved as to 2/1, ^ and y being treated as known, these Eqs. 
yield two resp. three roots, values of ?/, i.e., ordinates of the 
points where tangents resp. normals drawn through (x, y) meet 
the P ; hence, through any point may be drawn two tangents and 
three normals to P. The tangents are real and separate, real 
and coincident, or imaginary, according as 

2/^ — 4 gaj is > 0, = 0, or < ; 

i.e., according as the point from which they are drawn be with- 
out^ upon, or within the P. The sum of the roots is 2 y ; i.e., 



NORMALS TO P. 149 

the ordinate of the point through which the tangents are drawn 
is the half-sum of tlie ordioates of the points of touch ; i.e., the 
point is on the diameter of the chord of contact, as already 
known. 

The reduced Eq. of the normal is 

The absence of the term in yi shows that its coefficient, the 
sum of the roots, vanishes; or, 2/i' +2/i" +2/1'" = I i-^-, 
the sum of the ordinates of the points where three normals to a P, 
through a point, meet the P is ', or each is the negative sum of 
the other two. The sum of the ordinates of the ends of 1| 
chords is constant, namely, the double ordinate of their diam- 
eter ; hence all third normals through the intersections of pairs 
of normals at the ends of |1 chords cut the P at points hav- 
ing the same ?/i, i.e., at the same point; i.e., are the same 
normal; i.e., pairs of normals at the ends of \\ chords meet on 
a third normal. To find this normal, draw one of the |I chords 
through the vertex ; then the point symmetric as to the axis 
with the other end of the chord is the point of P through which 
the third normal goes. 

One of the normals is always real, wherever (a?, ?/) be taken ; 
the other two are real and separate, real and coincident, or 
imaginar}', according as ?/^ — 4gx< 0, = 0, or > ; i.e., 
according as the point they are drawn from is within, upon, or 
without the P. This is also clear geometricallv, since plainly 
real intersections of normals take place only within the 
curve. 

123. In the vertical Eq. of P, y^=4:qx, the principal 
parameter 4g is the double focal ordinate, or focal chord. In 
the Eq., y^ = 4:q'x, of P referred to any diameter and the 
tangent at its end, on putting x = q', the parameter 4 q' 
also appears as a double ordinate ; but is it also a focal chord ? 

Be p the focal ray to the origin or point of tangence, 0, o) the 
tangent's slope to the axis, p' the focal ray I| to the tangent 



150 



CO-ORDINATE GEOMETRY 



and sloped to, p" the counter-ra}- sloped tt + co ; then p = x'-j-q, 
/o'= 2 g : 1 — cosco, p" = 2q : 1 + cosw, p' -\- p"= 4zq : sin w. 
The second Eq. is got from the polar Eq. of the E by putting 
e = 1, = 7r — co; the third from the second by putting 
TT + o) for (0. Clearly p' + p" or -iq : sin co" is the focal chord |1 to 
the tangent ; the x of this chord is the focal distance of the inter- 
section of tangent and axis, i.e., the focal distance of the point of 




tangence, i.e., p. Now project p on the focal X to the tangent, 
and project this projection on the axis ; by Art. 121 the last pro- 

-2 



jection is g ; i.e. , p = q ■ sin co 



4: p — p' -{- p" = focal chord. 



Hence the abscissa to the focal ordinate is half that ordinate ; 
but this is the property of the abscissa q'. Hence 4 q' is the 

— ; 2 

double focal ordinate, or focal chord, and = 4 g : sin co , where 
(o is the axial angle or slope of the tangent. 



MAGIC EQUATIONS. 151 



CHAPTER VI. 

SPECIAL METHODS AND PROBLEMS. 
Magic Equations of Tangents and Normals. 

124. Thus far the Eq. of the tangent has been expressed 
through the Ccls. of the point of touch. This Eq. does not in 
itself determine the tangent, but only by help of an understood 
Eq. of condition declaring the point of toucli to be on the curve ; 
without this latter, it were the more general Eq. of a polar. 

Thus, y^y = '2q(xi-\-x) touches the Z' y^ = 4oqx only 
in case {Xi, y^) be on P, i.e., only in case yi= 4 qx^ ; other- 
wise, it is but the polar of (cCi, 2/1) as to the P 2/^ = 4 qx. This 
implied Eq. greatly cumbers operations about the tangent. 

Where not the point of tangence but onl}^ the direction of the 
tangent is involved, this cumbrance may be avoided by express- 
ing the Eq. of the tangent through the direction-coefficient s as 
the parameter of the Eq. This form of the Eq. of the tangent 
is called the magic equation of the tangent. 

It may be got by putting for y its value sx-{-d in the general 
Eq. of the conic, and expressing the condition of equal roots of 
the quadratic in x, whence may be found d in terms of s. But 
this is tedious. Better is it to get the special forms for £, //, P 
separately. Thus, after the above substitution in y^ — 4, qx, 
the roots are equal when s^d!'^ = {sd — 2 qY , or when d = q:s\ 

• • y = 8x-^- 

is the magic Eq. sought ; s being thought as changing, it is the 
Eq. of a family of RLs. touching the P 2/^ = 4 qx. 

In finding the magic Eq. for the £, we may exemplify another 
method. Solved as to y the ordinary Eq. is 



152 CO-OHDINATE GEOMETRY. 

y 

Here s 

square s, multiply by a^, add Ir ; results d? = s^o? -\- h^. 
Hence y = sx ± -y/s^d^ + ^^ 



1)' 


.^. 


,.+^^ 


0? 


2/1 


2/1 


6^ 
a' 


2/1 ' 


2/1 



resp. y = sx±-\/s^a^ — W 
is the magic Eq. for £ resp. /^. 

N.B. The steps in this elimination are suggested by the 
reflection that 

125. Magic Eqs. of normals are easy to find. Thus, in P^ 

s = — , 2/i = -^5 a?i = ^- 
2/i « 

Substituting in the Eq. of the normal 

1 

2/-2/i= -^(a^-^i)» 

1 

and writing s for , we get as Eq. sought 

gs^ + (2 g — cc) s + ?/ = 0. 



a2 



In £, y^ = y^ \ ■\/s^a^ + &^, 



whence cci = — s^a^ : Vs-a^ + 6^ ; 

whence, on substituting as in case of P, there results 

{y - sxY = s\h^ -a'Y: (sW^ + a^) 
as Eq. of normal to £. 

Changing b^ to — &^, we get a like Eq. of normal to H. 



THE ECCENTKIC ANGLE. 153 

These Eqs. are of fourth degree in s ; hence may be drawn 
from any point four normals to an £ or an /^. 

126. The use of the magic Eq. may be illustrated in finding 
the locus of the intersection of a pair of _L taugents to an £ : 

y = sx-\- ^s^o? 4- W and y = ic+-^— +6^. 

Clear, transpose, square, sum, and divide b}' 1 + s^ ; results ^ 
a? -{- y'^ = a^ -\- IP \ or, in case of //, a^ -j- ?/2 = cr — h^. 

These are named director-circles of £ resp. H. If £ and 
H be co-axial, the D.C. of each goes through the foci of the 
other. 

Show that the D.C. of P is the directrix. 

The Eccentric Angle. 

127. By Art. 107 the £ 

shadow or |1 projection of the circle 
projection chords I| to the plane of the £ are projected at full 
length, chords J_ to these are shortened in the ratio 6 : a, every 
other S3'stem of |1 chords are shortened in some ratio between 
1 and h : a. Clearl}' the centre of the circle is projected into 
the centre of the £, hence the diameters of the circle into the 
diameters of the £. A pair of _L diameters in the circle are 
conjugate, each halving all chords || to the other ; hence their 
projections are conjugate diameters of the £, each halving all 
chords II to the other. Call the diameter || to the plane of the 
£, which is projected into the axis major of the £, the axis of 
the circle ; then the ^ which any diameter makes with this axis 
is called the eccentric angle of the projection of that diameter. 
The eccentric ^ of a point of the £ is the eccentric ^ of the 
diameter through it. Hence the eccentric ^s of two conjugate 
diameters differ by 90°. 



0^ 


f 


= 1 


is the 


vertical 


a? 


h^ 










ircle 




^-^ + 2/^ 


= a\ 


In 


this 



154 



CO-ORDINATE GEOMETRY. 



If the projected circle be turned round its axis througli 



cos~^~, it will fall on the major 

circle of the £. If e be the eccen- 
tric ^ of (oj, y) on the £, then the ^/ 
Cds. {x\ y') of the corresponding 
point of the circle are 

a:;'=acosc, y'=asmc; 

hence x = a cos e, y =b sin g 

are the Eqs. of the £ in terms of e. 



128. The eccentric ^ is es[)eciall3- useful in dealing with 
chords and tangents. Thus the chord through e^, eg is 

= 0, 




X 


y 


1 


a cos q 


b sin €i 


1 


a cos €2 


b sin G2 


1 



which on reduction takes the form 



^2 



- COS " + ^ sm -^^ — - = cos — — 

a 2 6 2 2 



Putting e^ = 69 = e, we get the Eq. of the tangent at e : 

X ?/ 

- cos e + T sin e = 1 . 

a 

Replacing Xj^ and .Vi i" the Eq. of the normal, we get its Eq. 

ax by -,9 
^ = cr — b\ 

cos €i sin q 

The advantage of these Eqs. lies in the fact that the arbi- 
trary e's are free from condition. We may illusti-ate their use 
in finding the locus of the intersection of the pair of tangents 
at the ends of conjugate diameters. Such a pair are 



X 1/ 

- cos e + T sin e = 1 and 

a b 



X . y 

-sm e + -cos e= 1. 

a b 



QUASI-ECCENTRIC EQUATION OF H. 



155 



Finding hence cos e and sin e, and placing the sum of their 
squares equal to 1, we get, on reducing, 

9 9 

a co-axial E with axes multiplied by V2. 

129. We have seen that equiaxial H \^ to H in general as the 
circle (equiaxial E) is to £ in general. Any H may be thought 
as a vertical shadow or parallel projection of an equiaxial H 

under cos~^— (a>5). If a<.b, we may think the relation 
a 

reversed : the equiaxial H the projection of H in general. Now 

since sec 77 — tan7;''=l, if we put a; = a sec?/, as we 
may, we must have y = atiinr) in equiaxial H. or y = btanrj 
in H in general. These, then, are the Eqs. of H in terms of rj. 




To construct r} we have but to form a right A with base a and 
hypotenuse x ; the ^ at the base will be y. This is done by 
drawing from the end of the abscissa x a tangent to the major 
circle of the //. The tangent-length is the corresponding 2/ in 



a. 



the equiaxial H and the -th part of the y in the f/ in general. 

b 



156 CO-OEDINATE GEOxMETRY. 

The equiaxial //' 3/^ — a;^ = 1 , is got b}' exchanging x and 
y in the equiaxial H a? — y^=\'^ hence its Eqs. are 



In the general //' -^ — ■^=— 1, y {^ changed in the ratio 



a; = atan'>7, y^asecij 

fi ^ _ 1!. _ 
a' b 

- while X is unchanged ; hence the Eqs. of it are 
a 

x= a tan 7;, y = b sec 77. 

In the equiaxial H's conjugate diameters are equal and like- 
sloped, the one to the X-, the other to the y-axis ; hence the 
points corresponding to like values of 77 in the two pairs of Eqs. 
are ends of conjugate diameters ; after projection conjugate 
diameters remain conjugate, each still halving all chords || to 
the other ; hence like values of rj yield ends of conjugate diam- 
eters in the pairs of general Eqs. 

Noting the signs of the trigonometric functions, we see that 

7) ranging from to - yields all points in the right upper 

branches of H and //' ; 77 from - to tt yields all points on the 

left lower branches of both ; 77 from tt to -^ yields all on the 

left upper branch of H and the right lower branch of //' ; 77 

from — ^ to 2 TT yields all on the right lower branch of U and the 

left upper branch of W . Hence the ends of conjugate diame- 
ters, answering to like values of 7;, will be in the same quadrant 
for 77 between and tt, but in counter-quadrants for 77 between 
TT and 2 tt. Observing this,, we find as Eqs. of tangents through 
ends of conjugate diameters in first and third quadrants 

X y ^ x y 

- sec 77 — 7 tan 77 = 1 , - tan 77 — 7- sec 77 = — 1 : 

whence, on addition to eliminate 77, we get 

a 



HYPERBOLIC FUKCTIONS. 157 

i.e., one asj-mptote, as locus of the intersection of the tan- 
gents. 

Putting IT — 7] for -q in the second Eq., we get 

X ^ y 

tan v 4-- sec t? = — 1 

ah 

as Eq. of the tangent through the other end of the second diam- 
eter, whence 

a b 
i.e., the second asymptote, results as locus of the intersection. 

130. There is another noteworthy way of expressing the Cds. 
of a point on an ff through a third variable. As the student 
may know, sine and cosine may be defined analj'tically, without 
any geometric reference, through exponentials, thus : 

cos (9 = i (e*"^ + e-*^) , sin ^ = -^ (e^^ — e-'^) , 

where i • i = — l. 

All properties of sine and cosine may be drawn out from 
these definitions with greater ease and generality than from any 
other. 

If instead of i be written 1, the resulting expressions 

i(e0+e-e) and ^(e^ - e-^) 

are named resp. hyperholic cosine and sine of 0, and may be 

written licO and hs6. We see at once that hcO" — hsd = 1, 
and hence we may write in H 

- = hcOj ~ = hsd, or x= aJicO, y = hhs9, 
a h 

and in /^' x=ahs6, y = bhcO; the analog}' of which to the 
eccentric Eqs. of the £ is plain. Hyperbolic functions are of 
some use in higher analysis, and these Eqs. are of interest in 
Kinematic. 



158 



CO-OHDINATE GEOMETRY. 



Supplemental Chords. 

131. Two chords through a point of an £ or /^ and the ends 
of a diameter are called supplemental. Tliey are |1 to a pair of 
conjugate diameters^ for a diameter halving one is clearly |1 to 
the other. Hence, if on any diameter of the conic as a chord be 
described a circle-segment containing a given angle, and a point 
where this circle-segment cuts the conic be joined to the ends of 
the diameter, the diameters H to these chords will be conju- 
gate and inclined at the given angle. The problem of drawing 
conjugate diameters making a given ^ with each other is thus 
solved and soluble only when the circle-segment meets the 
conic in real points. As these points will in general be two, 
there are in general two pairs of conjugate diameters having a 
given slope to each other. 



Auxiliary Circles. 

132. Of these have already been found several, as : 

(1) and (2), the major and minor circles (Art. 107) ; the 
major is the locus of the foot 
of the focal _L on the tangent 
(Art. IIG). 

(3) The direct or -circle 

ar' + 2/' = tt' ± &' 

( + in £, — in H) , being the 
locus of the intersection of pairs 
of J_ tangents (Art. 125). 

(4) To these we may now 
add the two counter -circles. If 
either focal radius of any point 
of an E resp. H be lengthened 
resp. shortened by the length of 
the other, the point thus reached will clearly lie on a circle 
about the first focus, radius 2 a. Since the tano-ent halves 




AUXILIABY CIBCLES. 169 

the angle of the focal radii, the point and the other will be 
symmetric as to the tangent, or the point will be the counter- 
point of that focus as to the tangent. Hence the locus of the 
counter-point of either focus is the counter-circle about the otJiei' 
focus. Also, the counter-point of either focus (as to any tan- 
gent), the point of tangence, and the other focus lie on a RL. 

(5) The system of focal circles. Of these the X-axis is the 
common power-line, the Y-axis is the centre-line. Any one 
meets the tangents through its centre on the vertical tangents (at 
the ends of the major or real axis). For in the £ the radius of 
such a circle is v ^^ -^ 52 ^^^ ^-^ ^ being the eccentric ^ of the 
point of tangence of a tangent through its centre ; also the 
intercept of such a tangent between the vertical tangents is 

2'^ ct^ -{- b^ cot € . Like reasoning holds for the /^, cote chang- 
ing to QOSeCr], 

These circles are helpful in problems of construction. 

133. In P the minor circle lies wholly in 00 ; the major 
reduces to the vertical tangent, the locus of the foot of the focal 
_L on the tangent (Art. 121) ; the director-circle becomes the 
dii^ectrix, the locus of the intersection of pairs of _L tangents 
(Art. 106) ; so, too, does the counter-circle of the focus, since 
plainly the counter-points of the focus lie on the directrix (Art. 
121). All this the student may also prove analytically by pass- 
ing to the vertex as origin and reducing the Eqs. , remembering 

2 h^ 
that in Z', a = oc , e = 1 , — = 4 g. The focal circles all 

a 
reduce to the axis of P, since the other focus is at 00, and so 

are little useful in construction. 

Their place is filled in a measure by the circles ctbout As 
circumscribed about P, all of which pass through the focus, as 
may thus be proved. 

Let three tangents touch at Pi, P2, P3, and meet by twos at 
73, 7i, L ; by Art. 114, 

^F,FI, = ^I,FP„ and ^ P^FI^ = ^ I^FP, ; 



160 CO-OEDINATE GEOMETRY. 

whence ^ I^FI^ = ^ l.FPo + ^ P.^I, 

= \ -^{P.FI, + I,FP, + Ai^J, + I,FP,] 

^\^P^FP,^ 

i.e., the intercept of auy tangent between two fixed tangents to 
P subtends a fixed angle at the focus : lialf the ^ subtended by 
the chord through the fixed points oftangence. Now as the focal 
_L on the tangent meets it on the vertical tangent, the slope of 
the _L to the axis = the slope of the tangent to the vertical 




tangent; but by Art. 121 the former is half the slope of focal 
radius to the point of tangence ; hence the difference of the slopes 
of two tangents to the vertical tangent^ i.e., the ^ betiueen two 
tangents, is half the angle between the focal radii to the points of 
tangence. On apphing this to the case in hand, it appears that 
the ^s /i/o-^s and I^FI^ are supplementary ; hence the circle 
about the A I1I2T3 goes through F; q.e.d. Circles circumscrib- 
ing circumscribed As we may name focal. 

Vertical Equation of the Conic. 

134. It X — a resp. x + a be put for x in the central Eq. of 
the £ resp. //, the reduced Eq. takes the form 



LENGTHS OF TANGENTS. 161 

2/2= X -'Qi? resp. 1/" = x-\ — r-or, 

a a" a c(/ 

which is therefore the Eq. of the £ resp. H referred to the axis 

and the tangent through the left resp. right vertex. If in either 

2 52 
we put — . = 4 g, a= 00, we get the vertical Eq. of P 
a 

2 6^ 
?/2 z= 4 qx. Now 4 g or — is the parameter or double ordinate 

a 

through the focus ; hence these Eqs. state the geometric fact 
that the square of the ordinate^ as compared with the rectangle of 
'parameter and abscissa, shows : in the Ellipse, lack; in the 
Parabola, likeness; in the Hyperbola, excess. From this fact 
the curves seem to have been named. Hence the Eq. of any 
conic may be brought into the form y^ = Rx -\- /S'x^, where 
/S' is < 0, =0, > resp. for E, P, H resp. 



Lengths of Tangents from a Point to a Conic. 

135. These have been found to vary as the [1 diameters in 
the centric conic (Art. 101). In the P be Pi(iCi, 2/1), ^2(^2? 2/2) 
the points of tangence, ^1, ^2 the tangent-lengths. Then the Cds. 
of the intersection of the tangents are : 

dj — - — — V .//i.^25 y — — - — J 
4g 2 

i.e., are the geometric resp. arithmetic means of the Cds. of the 
points of tangence. Hence, pj and p2 being the focal radii, 



?/2 — y{ _L .^2 - Vi Vi _ 2/2 - 2/1" 



4 4g2 4 \^ q 

-2 



1+ 



4g 4g 



-2 



so 



h — p2 1 . . h ' h — Pi • P2 ? 

4g 



i.e., the squared tangent-lengths from a point to a P vary as the 
focal radii to the points of tangence. 



162 



CO-OEDINATE GEOMETRY. 



Areas of Segments and Sectors of a Conic. 

136. Bj' Art. 107 the area of an.£ is to the area of its major 
circle as b to a, chords _L to the axis major being all in this 
ratio. It is also clear that any segment of the £ is to the corre- 
sponding segment of the circle as b to a, and the same holds 
equally of half-segments reckoned from the axis major. 

Two corresponding sectors AOP, AOP' are made up of two 

half-segments in the ratio 6 : a, 
and two A in the same ratio ; 
hence are themselves in that 
ratio. So, too, are any other 
corresponding sectors AOQ^ 
AOQ' ; hence so, too, are their 
differences POQ, P'OQ'. 

Since the centric ^ ^ OP or 
(fi and the eccentric e or AOP' 
are connected by the relation 

y :x = tan </>=-• tan e, if the 

Cii 

sector be given by its centric ^, we may still use the 
eccentric. 

The area of au}' focal sector PFQ is the difference of the 
focal sectors AFQ and AFP^ each of which is, again, the dif- 
ference of a central sector and a A. 

137. In P be Pi, P/ any two points, I^ the pole of PjPi', M^ 
the mid-point. Then the RL. IiMi is a diameter, the tract IiMi is 
halved by P at jTi, the tangent at Tj is H to PiPi', and halves 
the tangents from 7i at I2, 1-2 - The As IiI^I^ and T^P^Pi have 
equal altitudes ; the base and therefore the area of the second is 
twice that of the first. "We may proceed with the As Tj/gPi, 
TiVPi exactly as with P^I^P^^ cutting off by mid-tangents at 
T^i T2 outer areas and by chords to T25 TJ inner areas twice as 
large ; and so on without end. The limit of the sum of the 
outer areas is the outer sector PiIiPi, and the limit of the sum 




AREAS OF SEGMEis^TS AND SECTORS OF A CONIC. 163 



of the inner areas is the inner parabolic segment P^T^P^^ cut 
off by the chord P^P^ ; hence this latter is twice the former, or 



of the A Pj^IiPi', or 



f of 




the parallelogram PiQi of the 
chord and the || tangent. 

The focal sector from the 
vertex V to the point P(x, y) 
is 

or \xy-\-^qy. 

If the axis and the diameter 
through P cut the directrix 
at Z>' and D, the area VD'DP 
is ixy-\-qy, i.e., twice the 
area of the focal sector. 
Hence, any focal sector PFP' 
is Jialf the area of the corre- 
sponding outer segment be- 
tween the curve and the directrix and the diameters through 
P.P'. 



138. If through any two points Pi, P^ on an M be drawn lis 
to either asymptote, meeting the other at Xi, Xg, then P^X^X^P^ 
is called an hyperbolic segment cor- 
responding to the hyperbolic sector 
PiOPs- The A PiOXi, Po^OX^ are 
equal, being halves of equal parallel- 
ograms (Art. 118) ; taking each in 
turn from the figure P1OX2P2, we get ^' 
in turn P1X1X2P2 and PiOXi ; hence, 
corresponding hyperbolic sector and 
segment are equal. 

If Pi, Qi and P2, Q2 be ends of two 
II chords, the. sectors P^OP^ and Q1OQ2 are equal ; for the con- 
jugate diameter of the chords halves both the triangular and 




164 



CO-ORDINATE GEOMETRY. 



hyperbolic areas, halving every element of each : taking away 
equals from equals, we have left the sectors, and therefore their 
corresponding segments, equal. 

If y = sx-\-b be any chord referred to the as3'mptotes, 

then the roots ccj, X2 of the Eq. xy = sof -\-bx = k^ are the a;'s 

2 

of the ends of the chord ; their product, , is independent of 

s 

5, i.e., is the same for all || chords. When the product of the 
x's is constant, so is the product of the y's by virtue of the rela- 
tion xy — K^. Hence either set of asymptotic Cds. of the 
ends of two |1 chords form a geometric progression. Plainly 
the converse holds: if any number of like asymptotic Cds., 
x's or 2/'s, be taken in geometric progression, they will belong 
to the ends of || chords, and hence the area of sector or seg- 
ment determined by any two consecutive ones will be constant. 
Take now the hyperbolic segment between the ordinates 
y = K and y = ci in the M xy = k^, 
and cut it into n equal sub-segments by 
ordinates in geometric progression. If rbe 




the ratio, then 



a = r"'K. 



and 



r = f - 



its base is — — k ; 
r 



The end-abscissae are x = k. x = — ■ . The 

a 

II sides of the first segment are k and tk ; 

its area, if ^ be the 

asymptotic angle, is <k[ — — k) sin </>. 

and > r/< ( — — K ) sin cf>. Hence the area S 
of the whole segment lies between 



K^sin<^n( 1) and k^ sm(fin(l — r) 



The ratio of these extremes is r ; if 71 be taken ever greater and 
greater, ?• nears 1, the extremes near each other, keeping S 
always between them ; hence S is the common limit which they 



HYPERBOLIC AEEAS. 165 

both near as n increases without Umit. We can readil}' evahi- 
ate n(l — r) or n -^ 1— (-j^ V for n nearing oo, by expanding 

- \n . To do this, we write it as a binomial, thus : f 1 -j 1 )« ; 



a 
as a is < K, 1 is < 1 ; accordingly, the binomial expansion 

K 

is applicable, being convergent. On expanding, taking from 1, 
and multiplying by n, there results the series, 

1 



n\l-{^t[=-\^-l+'l .^-1 



1 
1.2" 



i_,l.i_2 . i_i.l_2.l-3 

n n a -. , 'i^ n n 



1.2.3 K 1.2.3.4 



^-iV...|. 



As n rises above all limit, - sinks below all limit, nears 0, 

n 

and the numerators become —1, — 1.— 2, — 1.— 2.— 3, and 

the series becomes 



{ a ^ 1 a ^ . 1 a ^ 1 a ^ , } 
— J ^— T ^ +Q ■'■"7 ^ "^ C' 

(.K Zk ok 4k ) 

which, from Algebra, we know to be the negative of the expan- 
sion of the natural logarithm of - when 0< — = 2, as is the 

K K 

case here. Hence, at last, 

/8' = — K^.sin^ • nat. log-, or >iS' = K^sin<j5).nat. log— 

K a 

The area of a segment whose end-ordinates are a and &, 
6< a< K, 

= S(a, h) = K^ sin </> -j nat. log- — nat. log- >■ 
( b a ) 

= K^ sin cji . nat. log-. 
b 



166 CO-ORDINATE GEOMETRY. 

If K be taken as linear unit, and if ^ = 90°, i.e., if the H 
be equilateral, then 

/S'= nat. log-, /^(a, 5) = nat. log- ; 
a b 

i.e., the area of any segment, reckoned from the vertex, is the 
natural logarithm of the end-abscissa, and the area of any seg- 
ment wholly on one side of the vertex is the natural logarithm 
of the ratio of its end-ordinates. Hence natural logarithms 
have been called hyjperholic logarithms. 

Thus far, segments and sectors lie outside of H ; but problems 
about inner ones can now present no difficulty. 



Varieties of the Conic. 

139. We have found three species of conic : £, Z', H^ accord- 
ing as — C or h' — JiJ is < 0, =0, > ; but of each there are 
several varieties, which are now to classify. By Art. 51, on 
passing to H axes through a new origin (x-j, 2/1), the new abso- 
lute term becomes 

c' = (kx^ + Jiy^ + g)x^ + (Jix^ -j-jy^ +/)2/i + 9^i +/2/i + c. 

If the new origin {x^, 2/1) be the centre of the conic, the coeffi- 

cients of x-^^ y^ vanish, and the values of a^i, ?/i are — , — ; 
hence, 

Hence the Eq. of the centric conic referred to its centre is 

kx' + 2hxy+jf^-^^ = 0. 

Now, in the £, C or hj — h^i& > 0, hence Ic and j are like- 
signed, hence when A and they (A: and j) are unlike-signed the 
Eq. is clearly satisfied by real values of x and y, hence the £ is 
real ; but when A and they are like-signed the Eq. is satisfied 



VARIETIES OF THE CONIC. 



16T 



by no real values of x and 2/? hence the £ is imaginary ; also 
when A = no real values satisfy it but the pair (0, 0), 
hence the Eq. pictures a pair of imaginary E,Ls. intersecting in 
the origin (ci'i, 2/1) . 

In the /^, A: and J are unlike-signed and (7 < ; plainl}^ the 
Eq. is satisfied by real values of x and y in all cases, but for 
A = it pictures a pair of RLs. through the origin (xj, y^ . 

In case of the non-centric, P, (7=0, hence the first three 
terms of the general Eq. form a perfect square, and we have 

(■y/kx ± ■\/jyy-{-2gx-\-2fy-^c= 0. 
Solved as to the parenthesis, this Eq. takes the form 



V^ • X ± -y/j ' y -f 



/ _ 



± — = V2 (^a^ - iT. 



Accordingly, the Eq. pictures a real P save when (x = 0. 
Then it pictures two RLs., which are ahcays \\, and are real 
and separate, coincident, or imaginar}', according as K<i 0, 
= 0, or >0. If i = 0, like conclusions hold on changing 
6r to jP and K to J. 



Hence the foUowiuo- table 



O>0 



= 



C<0 




A.-A<0 

A = 

M>0 



real ellipse. 

two imaginary RLs. 

imaginary ellipse. 

real parabola. 
two parallel RLs. 
real parabola. 

real liyperbola. 
two real RLs. 
real hyperbola. 



168 CO-OKDINATE GEOMETKY. 



CHAPTER yil. 

SPECIAL METHODS AND PROBLEMS (Continued). 
Determination and Construction of the Conic. 

140. The Eq. of the conic has six constants, but division by 
any one reduces the number to five, which are therefore the 
independent arbitraries of the Eq. Hence five independent 
simple conditions are needed and enough to determine a conic, 
since they determine the five arbitraries. Conditions are inde- 
pendent when no one can be drawn out from the others, simple 
when each fixes but one relation among the arbitraries. Such 
are that the conic shall go through certain points or touch cer- 
tain RLs. A midtiple condition fixes more than one relation 
among the arbitraries. Thus, that the conic toucJi a certain RL. 
at a certain point is a double condition, namel}^, that the conic 
pass through two given consecutive points ; that the centre be 
(^15 2/i) is a double condition, for it makes 

kx^ + 7i?/i + ^ = and hx^ +jyi -\-f= ; 

that a given direction be asymptotic is simple^ since it fixes one 
point at CO, but that a given RL. be an asymptote is double, 
since it fixes two points at cc ; that a given point be focus is 
double, since three tangents besides would determine the conic 
by determining three points of the major circle ; to give the 

direction of an axis is to give a point (at oo ) in P, or to give 

2 h 
the relation = a constant in £ and //, a simple condition ; 

k-j 

hence, to give both axes in position, since this fixes the centre 

besides, is to impose a triple condition. A given eccentricity is 

in general one simple condition, but if e = 0, the conic is a 



SYSTEMS OF CONICS. 169 

circle, which implies the two conditions of Art. 58. For exer- 
cises, see pp. 210-213. 

141. The case of a conic fixed by five points merits special 
attention. If the five lie on a RL., that RL. and an}' other are 
the conic ; there is a two-told go of solutions in pairs of RLs. 
If only four lie on a RL., that RL. and any other through the 
fifth are the conic ; there is a 07ie-to\d qo of solutions in pairs of 
RLs. If only three lie on a RL., that RL. and the RL. through 
the other two are the conic ; there is but one solution. Thus 
far the solutions have been pairs of RLs., since no curve-conic 
has three points on a RL. If only two points lie on a RL., the 
Eq. of the conic is got by assuming 

Icx^ + 2 hxy -\-jy^ -{-2 gx -\- 2fy -f- c = 0, 

which with five like Eqs. got by replacing the current Cds. by 
the Cds. of the five points will form a system of six Eqs. homo- 
geneous of first degree in the six unknowns, k, h, J, g, /, c ; 
since the absolutes are all 0, the condition of consistence is that 
the determinant of the coefiflcients of the unknowns vanish ; 
hence that determinant equated to is the Eq. sought. But to 
avoid the tedium of reducing this determinant, we may proceed 
better thus : 

Be Li2 = 0, X23 = 0, X34=0, X4i = 0, Xi3 = 0, ^24 = 
the six RLs. fixed by four of the points ; to denote that in any 
L the current Cds. are replaced by the Cds. of the fifth point, 
prefix the superscript 5. Then 

Xi2-X34 = and i23-Ai = 

are a pair of pairs of RLs., i.e., a pair of conies through the 
four points; hence Li2'L^a~^L2z' L^i=-0 is a conic, being 
of second degree, through the four points. If this conic goes 
through the fifth point, then ^ii2- ^^34 — A^Z/23'^^4i = ; 
whence finding the value of A and putting it for X in the other 
Eq. we get the Eq. of the conic sought. 

Since we can find a fit value of X for any fifth point, it follows 
that Xi2'-^34 — '^-^28* Ai = represents a conic passing 



170 CO-OEDINATE GEOMETRY. 

through the four points and any other point. Also a conic 
through four points of which no three are on a RL. can sustain 
but one more condition, and A. may be taken to satisfy any one 
condition ; hence the above Eq. represents the whole sj'stem of 
conies through four points ; and since for any fiftli point there 
is found but one A^alue of A, it is seen that through Jive points^ 
no three of whicii lie on a RL., one, and only one, conic passes. 

142. Two special forms of this Eq. of a system of conies are 
speciall}' useful. 

(1) Take two of the Z/'s for axes ; then their Eqs. are x = 0, 
y= ; the Eqs. of the other two are 

Ix -\- my -f- 1 = and I'x + m'y -1-1 = 0, 
and the Eq. of the system is 

(Ix -\- my + 1) {I'x 4- m'y -\-l) — Xxy = 0. 
The terms of second degree are 

ll'x^ -f- {Im' -\- I'm — X)xy-{- 7nm'y^. 

They form a square, i.e., the conic is a P, when, and only 
when, 

{Im' -^ I'm - xy = AU'mm' . 

This Eq. is of second degree in A ; hence two, and only tiuo, of 
a system of conies through four points are P's. The student may 
easily investigate the conditions under which the P's are real or 
imaginary, separate or coincident. 

(2) Take the Eqs. of the RLs. in the N. F., and drop the sec- 
ond subscript ; then N^- JVg — AA^2 -^4 = is a conic about 
the 4-side whose counter-sides are JV^ = 0, -^3 = and 
JSF2 = 0, JV4=0. Call the lengths of the sides, i.e., the 
chords of the conic, Ci, Cg, C3, c^. From any point P draw rays 
I'l^ '^25 ^3? ^'47 to the vertices of the 4-side. Then the double 
areas of the As are 



whence 



CONIC THKOUGH FIVE POINTS. 171 



JVi • Ci = Ti ' T2 ' sin iVa, JSF^' C2 = r2' i\ • sin 7^*35 

^3 . C3 = rg • r4 • sin r3?'4, ^4 • C4 = 7-4 • ri • sin r^r-^ ; 

iVi • ^3 Ci • Cg _ sin 7V'2 • sin r37'4 



^2 • -^4 C2 • C4 



sin ?v 3 * sin r4?'i 




The right member of this Eq. is the cross-ratio of the 
four rays, and the left is constant so long as Ni'N^-. N^' -^4 is ; 
i.e., so long as the centre of the pencil, P, is on a conic through 
the four points. Hence the cross-ratio of a pencil from any 
point of a conic through four fixed points of the conic is con- 
stant. 

143. If now we take any sixth point on the conic, we shall 
have an inscribed 6-side. The sides and vertices numbered 1 
and 4, 2 and 5, 3 and 6, i.e., whose numbers differ by 3, are 
called counter. Let the Eqs. of the sides of the primary 
4-side stand as in Art. 142 ; then the Eq. of the fourth side of 
the 6-side, i.e., the side from 4 to 5, will be .A^g— kJV4 = 0, 
since it is a ray of the pencil (^3, N^ ; so, too, the sixth side, 
from 6 to 1, will have for its Eq. JVi — k'N^ = 0. Since 5 is 
on the conic iViiVg — XN^N^ = 0, we have 



172 



CO-OKDINATE GEOMETRY. 



hence the ray from 2 to 5 is k2^i — XI^2 = 0. By hke rea- 

soning, the ray from 3 to 6 is k'JSF^ — XJSf2 = 0. Hence the 
fifth side is the common ray of the two pencils 




These Eq's. are the same when fi=l : k\ /x' = 1 : k ; hence 
the fifth side is KiVj - AiV2 + K'JVg- /ck'^4 = 0. The RL. 
through the intersections of counter sides ^i = and 
N^ — kN^^ = 0, ^3 = and N-^ — k'N^^ =0 is the common 
ray of the pencils 

iVi - r (JV3 - kN^) = 0, N^- k'JST^ - r'JSTs = 0, 

i.e., it is the RL. kN^ — kk'N^ + k^N^ = 0. But this RL. goes 
through the intersection of the third pair of counter sides 
iVa = and kN^ — kk'N^^ -f- kN^ — XN^ = 0. Hence we have 

Pascal's Theorem. The three pairs of counter sides of a Q-side 
inscribed in a conic intersect on a RL. called a Pascal (RL.). 
Since six points may be thought arranged circularly in 5 ! or 
120 ways, and since a ray joining two points may be counted in 
two ways, it follows that there are sixty really different orders, 
and so sixty really different Pascal RLs. 



pascal's theorem. 



173 



Pascal discovered this beautiful relation, and built upon it a 
theory of the conic. 



144. We have seen (Art. 141) how to form the Eq. of a 
conic through five points ; Pascal's Theorem enables us to con- 
struct it without knowing its Eq. , and more, rapidly than by 
Art. 71, thus : 

Draw through 5 any RL. ; it will cut the conic in 6 ; find the 
intersections of sides 1 and 4, 
2 and 5 (the ray just drawn) ; 
through them draw the Pascal 
RL. ; it will pass through the 
intersection of sides 3 and 6 ; 
from 1 draw side 6 through 
the intersection of the Pascal 
and side 3 ; it will meet side 5 
in point 6. So we may find 
any number of points of the 
conic. 

If side 5 be drawn |1 to side 
1, the RL. halving the tracts 

1 2 and 5 6 will be a diameter of the conic ; a second diameter 
can be got by drawing side 5 H to side 3 ; these two diameters 
fix the conic's centre. 

If side 5 is to touch the conic at 5, point 6 must fall on 5 ; 
hence, draw side 6 from 1 through 5 ; through the intersections 
of the counter-sides 1 and 4, 3 and 6, draw the Pascal ; it will 
cut side 2 in a point of side 5 ; the RL. through this point and 
5 will touch the conic at 5. So we may draw any number of 
tangents to the conic. 

145. We have learned to consti'uct the conic by points and 
form its Eq. from given conditions ; it remains to determine its 
elements, centre^ axes, foci, directrices, asymptotes, from its Eq. 

By Art. 93 the centre is the point {G:C, F:C). If O be 
> resp. < 0, the Cds. are finite, the conic is an E resp. H. 




174 CO-OKDINATE GEOMETRY. 

Transformed to H axes through the centre, the Eq. becomes 

G 
The asymptotes are now 

TcQi? -\- 2 hxy -\-jy^ = ; 

hence, if we pass back to the old origin and axes, the Eq. of the 
asymptotes becomes 

kx" + 27ixy -\-jy^ J^2gx + 2fy + g — -— = 0; 

i.e., the Eqs. of the conic and its asymptotes differ only by the 

constant -— 
G 

The axes halve the ^s between the asymptotes ; hence they 

are the RLs. x^ — y^= ~^ xy by Art. 53. As tan 20 = , 

h _ k -j 

$ being the slope of an axis to the X-axis, we may now find the 

2h 

directions of the axes by drawing the RL. y = x and halv- 

k-j 

ing its slope to the X-axis ; but to determine the lengths of the 
axes is still tedious. A guide to a simpler solution of both 
problems in one is the reflection that the diameters of a conic 
through its intersections with a concentric circle are equal, 
being diameters of the circle, hence are like-sloped to either 
axis of the conic ; accordingly, when these diameters fall 
together, it is on an axis of the conic. Now by Arts. 30, 50 

kx^+2hxy+jy + ^.^±t = o 

is a pair of RLs. through the intersections of the conic with the 
concentric circle x^-{-y^ = r^, and the origin (centre). They 
fall together when the Eq. is a square, i.e., when 



ECCENTRICITIES. 175 

The roots r}^ ri of this Eq. are therefore the squared half- 
axes of the conic ; putting them in turn for r^ in the Eq. of the 
pair and taking the second root, we get the Eqs. of the axes. 
The axes thus found in size and position, the foci are found by 
laying off on the proper axis the proper focal distance, 
ae = Va^^~6^. The directrices are easily constructed as polars 
of the foci. 

To find the eccentricity, suppose the Eq. of the curve referred 
to its axes to be Ax^ -\- By^ = 1 ; then, as A and B are the 
reciprocals of the squared half-axes, we have A = B(1 — e^). 
Now pass back to the original central Eq. 

kx^ + 2 hxy -\-jy^ -f — = 0. 
O 

By Art. 102, since o> = 90°, 

A-\-B = k+j and AB = kj — hK 
On elimination of A and B from the three Eqs., there results 

e^^(^^^jy±^(e^-l) = 0. 
kj-h' ^ ^ 

This Eq. teaches that there are two pairs of counter-eccen- 
tricities of a centric conic, corresponding to the two pairs of 
foci, one real, one imaginary ; if e^^, eg^ be the squares, 

2 I 2 2 2 (k — jY -{- Ah^ 

' kj-h^ 

Hence — - H — ^ = 1 5 the sum of the reciprocals of the squared 

eccentricities is 1. Also, if kj — h^ > 0, the product of e^ and ei 
is — ; i.e., the squared eccentricities are one -f-5 one — ; i.e., 
in E one pair of eccentricities are real, one imaginar3\ If 
kj — h^<cO, both the sum and the product of the squared 
eccentricities are -f- ; hence both squares are +, hence in H 
both pairs of eccentricities are real. The real foci lie on the real 
axis, which latter falls on the X- resp. F-axis when A resp. B 
is -f, and the corresponding squared eccentricity is 



176 CO-ORDINATE GEOMETRY. 

B-A A~B 

resp. — • 

B ^ A 

146. The Eq. of /*, as the highest terms form a square, may 
be written 

The RL. kx-\- ly = is a diameter (Art. 97) , and the RL. 
2gx-\- 2fy -f (7= is a tangent at its end, since on combining 
the last Eq. with the Eq. of the P there result two equal pairs 
of values of x and y^ which also satisfy KX-\-iy=Q. This 
diameter and tangent are not in general _L, since not in general 

is ^^ = — 1. But by adding and subtracting 2 kA + 2 tX + A.^ 

we may write the Eq. of the P thus : 

{KX-^iy-\-\y + 2{g--KX)x-^2{f-i\)y + c-X'=0. 

The RL. Kic + ty + A = is still a diameter, being || to 
K£c + t?/ = 0, and 2{g — KX)x-\-2{f— LX)y -\- c — X^ =0 is the 
tangent at its end for the same reason as before ; they are _L if 

— - = •^~ ^ , i.e., when, and only when, X = ^^ Y ' 
I g — kX k^ + t 

Hence when X has this value, the two RLs. are resp. axis and 
vertical tangent of the P, and their intersection is the vertex. 

The parameter 4 q is the ratio of the squared distance of a 
point of P from the axis to its distance from the vertical tan- 
gent ; i.e., 

lg^ (^^ + ^y + X)\ 2{g-KX)x-\-2{f-iX)y + c-X'^ 



^ 2V{g-KXy + (f-^0^' 
k' + l' 

whence on substitution for X, 

1^^ 2(KQr-/) ^ 2(V^.7-V//) 



TRACING CONIC S. 177 

The exponent f leaves the sign of 4g at will, as it should be, 
since the + direction of the X-axis is 3'et at will. The /? 
clearly lies on the — side of 

2(g- KX)x-t 2(/- LX)y + c - A^ = 0.' 

The focus is on the axis q distant from the vertex ; or we 
may determine its Cds. by finding the intersection of the tangent 
2gx-\- 2fy + c = and the vertical tangent, and through it 
drawing a _L to the former, which will cut the axis at the focus. 

147. When the elements are found, the curve may be actually 
traced by the methods of 2)oints^ of tangents^ or of continuous 
motion. 

(1) Cut the major axis of an E anywhere within the foci ; 
with the two parts as radii draw circles about the foci ; each 
pair of circles, the sum of whose radii is the axis major, inter- 
sect in two points of the E (Art. 101). Like holds for the H 
on changing within to without and sum to difference. 

(2) From the end of any radius of the major circle of an £, 
drop a _L on the axis major ; from the intersection of the radius 
with the minor circle draw a || to the axis major ; it will cut 
the J- in a point of the £ ; for it cuts it in the ratio b : a. 

From any point on the real axis of an H draw a tangent- 
tract to the major circle ; also draw a jl tangent-tract to the 
minor circle ; at the point lay off the equal of the second tan- 
gent-tract _L to the axis ; its end is a point of the // ; for the 
first tangent-tract is clearly a tan?;, and the second is 6 tan 77 
or y. 

(3) Through a focus draw chords of the major circle of the 
£ resp. H ; through their ends draw _Ls to them ; the ±s touch £ 
resp. // ; for the feet of focal ±s on the tangents lie on the 
major circle. 

(4) Join cross-wise the intersections of a focal circle with the 
vertical tangents to an £ resp. // ; the junction-lines will touch 
the £ resp. //, by Art. 132. 



178 



CO-ORDINATE GEOMETRY. 



By either (3) or (4), more easily by (4), enough tangents 
may soon be drawn to shadow forth the curve quite clearly. 

(5) If the ends of a string 2 a long be fastened less than 2 a 
apart, and it be stretched by a sliding pencil, this will trace an 
£ whose axis major is the lengthy whose /oa are the ends^ of the 
string. 

(5') If the ends of a string be fastened, one at a point, the 
other at the end of a ruler 2 a longer than the string, and the 
string be kept stretched against the ruler by a pencil while the 
ruler turns about its other end fastened at a second point more 
than 2 a apart from the first, the pencil-point will move on an H 
whose foci are the fixed points and whose real axis is 2 a. 

These constructions rest on the same properties as those of 

(6) If one end JV of a ruler a long slide on a fixed bar BB^ 

while a fixed point N' of the 
ruler slides along a JL bar AA\ 
the other end P of the ruler will 
trace an £ whose axis major is 
2 NP= 2 a, whose axis minor is 
2N'P= 2 h. For draw a circle 
with radius a about the cross- 
point (7, and a radius CP' \\ to 
NP. Then MP : MP^ = b:a. 
The three bars form a pair of 
elliptic compasses. 

(6^) If a ruler RBW bent at B slide 
along a straight-edge DD' while a pencil- 
point P keeps a string RB long stretched 
against the ruler, one end of the string being 
F fastened at i?, the other at a fixed point F^ 
then P will trace an H^ of which P is a focus, 
DU a directrix, BR an asymptotic direction. 
For the distances of P from F and DD' are 
clearly in a fixed ratio, namely, sec ^, where 







~~~^^^ 


P' 


fx- 


B 


^ 


p\ 


A' 


C 


// 


^^ 


\ 




A 


mJ 


K 




/ 


yj 


\ 


B' 


^ 


/ 




CONFOCAL CONICS. 179 

e = XEBR'-90°; hence e = sec^, l = cos^, and cos-^l 

e e 

is the slope of an asymptote to the X-axis. 

Among constructions of P by points this seems simplest : 

(7) About the focus with an}'' radius >q draw a circle ; from 
where it cuts the axis lay off 2 g toward the focus ; through the 
point thus reached draw a _L to the axis ; it will cut the circle in 
points of F ; also the junction-lines of the points of F and the 
other end of the diameter will touch the P (Art. 121). 

(8) From the focus draw any ray to the vertical tangent ; 
through their intersection draw a X to the ray ; it will touch P ; 
also a second focal ray having twice the slope of the first to the 
axis will meet the tangent at the point of tangence (Art. 121). 

Or, the point of touch maj' be found by remembering that the 
focal _L on the tangent halves the tangent-tract from the axis. 
Constructions of P by points and by tangents involve each other. 

(9) Construction (6') of //will yield P when the ruler is bent 
at right angles ; for then ^ = 0, e = 1. 

Confocal Conies. 

148. Confocal conies are clearly also co-axial, for the foci fix 
the axes in position ; if 2 a, 2 6 and 2 ai, 2&, be their axes, then 
a^ —b^ = a^ — bi — squared central distance of focus. 

Conversely, ?*/ itoo conies he co-axial and a^ — b^ = ai^ — bi, 
they are confocal, the foci clearl}' falling together. 

On transposing, we get a^ — ai =b^ — bi ; accordingly we 
may put a^ -\-X for ai, at the same time putting 5^ + X for b^^. 

9 9 

X 1/ 

Hence — h^r^ — = 1 is a family of confocal conies, 

a^ + A 6--f-A "^ 

and all conies confocal with the base conic f- ^ = 1 are 

a' b' 
got by letting X range from -co to -f go. 

For — GO < A< — a^, the conies are imaginary ; for A = — a^, 

the conic is a pair of RLs. fallen together on the F-axis ; for 



180 CO-ORDINATE GEOMETRY. 

— a^<X< — 6^, the conies are //'s sinking down to the X-axis ; 
for \= —b'^— 0, the conic is the doubly-laid X-axis tvUJiout 
the foci ; for A = — 6^ + 0, it is the doubly -laid X-axis 
ivitliin the foci; for — 6^<A< oo, the conies are £'s swelling 
out from the X-axis toward the concentric circle with oo radius. 

The Eq. — h -f- 1 = is of second degree in X : 

a^-f A ¥-\-X ° 

hence to any pair (x,?/) correspond two values of A ; i.e., through 
any point in the plane ^oass tivo, and only two, confocals. For A 
very great and + , the quadratic is — ; for A nearing — 6^, it is 
+ ; for A = — &^, it springs from + oo to — oo ; for A nearing 

— a^, it is again + ; for A = — a^, it again springs from -f- oo 
to —00, and thence stays — . Accordingly, the quadratic 
changes sign by passing through only for A between + oo and 

— 6^, and for A between — b^ and — a^ ; i.e., the two confocals 
which pass through any point are the one an £, the other an H. 

The tangent and normal of the £ halve the outer resp. inner 
!^s of the focal radii to the point, the normal and tangent of the 
H halve the same ^s ; hence they are the same pair of RLs., the 
normal of one curve is the tangent of the other ; hence confocals 
cut each other, loherever they cut, orthogonally. 

149. Confocal curves (and especially confocal surfaces) form 
a system of rectang. Cd. lines (and surfaces) of great import to 
Higher Geometry and Mechanics. It is in place here only to 
show how ordinary Cds. may be expressed through confocal 
Cds. : pairs of values of A yielding confocals through the point. 

Be _^_4-_^-=l and , ^' + -^^ = 1 
a^ + Ai &' + Ai a^ + As b'-j-X^ 

the confocals. 

Multiply by the divisors of y^ and subtract ; there results 



(. a" -f Ai a^ + Ao J 



a" -f Ai a^ + Ao 

= (Ai - As) (a' + A,) {a' + Ag) : (A^ - Ao) (a' - 6-) 

^ (a- + X0(a^ + A2) 
a'-b' 



SIMILAR CONICS. 181 



and on exchanging or and 5 , 



y ,2 7.2 



Hence a? -[- y'- = a" + ly' + X^ + X2 = o? ^- K^-^' + ^2 



= a? + X. + y' ^ X^ = r\ 

This r being regarded as a half -diameter of the confocal Ai, 
its conjugate ?\' is given by the Eq. 



ri'2 = ^2 + Ai + 5' + Ai - a^ + Ai + 6" + A2 

= Ai — A2 j 

and so ^2'^ = ''^2 — Ai; 

whence r^' ^ + rg' ^ = ; 

i.e., the conjugates to a common diameter of two confocals are 
alike in size, one real, one imaginary. Also, by Art. Ill, if ^1 
resp. P2 t>e the central _L on the tangent to Ai resp. A2 H to r/ 
resp. rg', or through the end of r, then 

Aj — A2 A2 — Aj 

Thus may all geometric elements of the sys.tem of confocals 
be expressed through the parameters Aj, A2, and their relations 
studied. 

Similar Conies. 

150. From Art. 82 it is clear that any conic K^ similar to the 
central conic K must be congruent with some conic K^ concen- 
tric with K and similar to it ; K^ and K2 differ only in that K^ 
has been X)^^slied and turned as to K. Since the centres of K 
and K2 fall together in the centre of similitude, the centres of K 
and Ki are corresponding points or centres of similarity. 

As K^ and Ko are congruent, in dealing with their metric 
relations we may put the one for the other. Now the eccen- 
tricity of ^is la^-Jy" : a^J^, that of K^ or K^ is \a^^ - \^ : a^^\h', 



182 CO-ORDINATE GEOMETRY, 

in K and /12, a and ag (or a^) correspond, so do h and h^ ; 

hence, 

ah a a 



or 



U/J 



tti hi h hi^ 

hence the eccentricities of similar conies are equal. Conversely, 



if 6=^2 = ^1, 


or if 




a'-h'' _ a,2 _ h^ 




0? ai 


by decomposing. 


we get 




a h 

CI2 &2 ' 


or the conies are 


similar. 



Hence eccentricities equal is the necessary and sufficient condi- 
tion of similarity in central conies. 

As P is simply a central conic whose centre has retired to 00 
while the parameter has stayed finite, we may at once infer that 
all P's are similar, having the same eccentricity e = 1, and all 
conies similar to a Z' are P's. Or we may place the P's vertex 

on vertex, axis on axis; then their Eqs, will be y^ = 4:qx, 

2 2 

y^ = 4:qiX; or ps'inO =^ 4: qp cos 6, p^sinOi = 4: q^p^ cos O^ ; 

hence for = 0^^ p- pi = 9 '9ii or the P's are similar, the ratio 
of similitude being the ratio of their parameters. 

Since in similar conies the eccentricities are equal, the expres- 
sion (2 — e^) : 1 — e^ must be the same in them ; hence, by Art. 
145, (k-{-jy : kj — h^ must be constant and equal 



—2 



This, then, is the condition that K and /r^, given hy their 
Eqs., he similar. For oblique axes we must write 

k -f i — 2h costo 

for k-\-j, as is plain from Art. 102. 

If now K and K^ be like-j)laced, i.e., have corresponding 
tracts II, the Eq. of K2 becomes that of Ki on change of origin 



CENTBAL PliOJECTION. 183 

only ; by such change Z^i, Z^i, j\ are not changed, hence they are 
the same in the Eqs. of Ki and K2 1 hut in K and /ig the inter- 
cepts on the axes, oi'igin being at centre, vary inversely as VA;, 
VA;i (or '\/k2), Vj, Vji ; and since these tracts correspond, 

Jc : ki =j :ji = 7^ ; and since ^^ — 'TJl. is constant, we have also 

In two P's like-placed, or with 1| axes, it is plain that 

Jc Tx 
— = -^, since each is the squared direction-coefficient of the 

J ji 

axis ; hence in all cases the triple equality — = - — =^ shows 

that the two conies are similar and similarly placed. 



The Conic as the Projection of a Circle. 

151. By Art. 107 the £ is a'|| projection of a ciTcle, the 
only kind of projection yet spoken of. The notion may be 
widened thus : ^ 

The point P' ivhere any plane 11' cuts any ray from S is called 
the centYsl p>rojection of any point P of the ray^ on the plane. S 
is called the centre of projection, n' the plane of projection. 
Only when /S is at oo does central pass over into H projec- 
tion. 

Rays from S through points on a RL. lie in a plane ; hence 
their intersections with n' lie on a RL. ; i.e., the projection of a 
RL. is a RL. 

If this RL. cut a plane curve in n points, its projection will 
cut the projection of the curve in n points, the projections of the 
original n points ; since n fixes the degree of the curve, that 
degree is unchanged by j^TOJection ; e.g., the projection of a conic 
is a conic. 

If the points in which the RL. meets the curve be consecu- 
tive, so will be their projections ; hence, the projection of a tan- 
gent to a curve is tangent to the projection of the curve. 



184 CO-ORDINATE GEOMETKY. 

Hence, too, i\\Q projections of pole and po?ar as to a conic are 
X)ole and po/cfr as to the jyi'oject ion of the conic ; also, projections 
of conjugates^ points or RL's., are conjugate. 

All points of a plane through the centre || to 11', and no 
others, are projected into oo ; hence, to project a RL. into go, 
p)roject it on a plane \\ to the plane through it and the centre. 

A p)encil of rays is clearly projected into a p)encil of rays, 
whose centre is the projection of the centre of the pencil. This 
latter projection will be in Jinity unless the centre lie on a RL., 
the intersection of the projected plane 11 and the plane 11" 
through the centre >S', || to U' ; then it is in go. Hence any 
pencil whose centre is on this RL., IL'\ is projected into a pen- 
cil of II RLs. The centres of all such pencils lay on a RL. 
])efore projection ; hence they lie on a RL. after projection, 
namely, the RL. at go. 

The plane 11'" through the centre jS \\ to IT meets n in a RL. 
at 00, and meets TI' in a RL. in finity, I"'L' ; hence all pencils 
of II MLs. in n are projected into pencils of intersecting RLs. 
in n', whose centres lie 07i the RL. I"'L'. 

152. Since |1 planes meet a third plane in |1 RLs., and 
since ^s between H pairs of (like-directed) RLs. are equal, 
clearly any ^ in 11 is protected into equal ^s 07i || planes. 
Hence, if the sides of an}- !^ at ^ cut ZL" at B and C, then the 
projection ^5'^'(7' of ^^^Con n' equals ^BSC, the projec- 
tion of the same ^ on 11". Hence, to project any ^ BA^G 
into an ^ ai, draw BS and CS making <BSC = ^a^, then 
project on any plane || to the plane BSC This may be done 
in an go of ways. If some RL. of the plane of BA^C is to be 
projected into go, let it cut the sides of the ^ at A-^ in ^i, C^ ; on 
jBiCi as chord draw in any plane an arc to contain the given ^ 
ci ; the centre S may be taken anywhere on this arc, and 11' 
may be any plane || to the plane B^SCi. If i'^^o ^s in a plane, 
at Ai and Ao, are to be projected into tivo given ^s, a^ and a^, 
while a given RL. in the plane is to be projected into go, con- 
struct as before in any plane on BiCi as chord an arc to contain 



OKIGIX 01 THE X^^ZSIE COXIC. 



185 



tti, and in the same plane on B.2C.2 as chord an arc to contain o.., : 
the intersection of these arcs determines the centre aS, which, 
however, may still lie anywhere on the circle of intersection of 
the surfaces generated by revolving the arcs about their chords. 
The plane of projection 11' ma}' be any plane 1| to the plane 
thi'ough the centre aS and the give A EL. 

On this theorem, that any two angles in a plane may he pro- 
jected into angles given t?i size and at the same time a given EL. 
of the plane p)roJected into x, is based the theory of prrojections. 
An immediate deduction is : 



153. Any conic may be projected into a circle and at the same 
time any point into the centre of the circle. 



^^ 




Be C the point to be projected into the centre of the circle. 
Let the polars of any two points Pj, Po on the polar of C as to 
the conic cut that polar in Qi, Q2 '■> then are P^, Qi and Po, Q., 
pairs of conjugate points. By Art. 152 project '^P-JJQ^ and 
^PoCQo each into a P^ and the polar of C into x ; then is C 
the centi'e of the projection of the conic (which is itself a conic, 
by Art. 151). since its polar is at x; also Pi'C", Qi'C and 
P2C'. QoC are tico pairs of conjugate diameters Sit E^s. Hence 
the projection is a circle. 



186 CO-ORDINATE GEOMETRY. 

If now we hold the centre of projection fast, and exchange n 
and n', we shall get the correlate theorem : 

A circle may be projected into an}' conic, and at the same 
time the centre of the circle into any point of the conic. 

The system of rays from S to the points of the circle form a 
circular cone; the projectiorf of the circle on any plane is the 
intersection of that cone and the plane; hence, any conic is the 
intersection of a plane and a circular cone, and any intersection 
of a plane and a circular cone is a conic. Hence the name 
conic section or conic. 

The student will now readily see that the section of a right 
circular cone is a circle when the cutting plane is J_ to the axis 
of the cone ; as the plane turns, the section passes over into an 
£ with increasing eccentricit}' till the plane gets H to the edge 
of the cone, when the section becomes a Z' ; as the plane still 
turns, the section becomes an H. 

Again, if a circle stand upright on a plane, and a centre of 
projection descend toward the plane, the projection of the circle 
on the plane will be an E till the centre reaches the level of the 
highest point of the circle, when it becomes a /* ; as the centre 
still descends, the projection becomes an H ; as the centre passes 
through the plane, the H shrinks to a pair of RLs. fallen 
together, and swells again into an £ as the centre sinks below 
the plane. 

Properties of a curve not changed by projection are called 
projective properties. By Art. 41 properties connected with the 
cross-ratio are such. But this thought cannot be followed up 
here. 



HOMOGENEOUS CO-OKDINATES. 187 



CHAPTER VIII. 
THE CONIC AS ENVELOPE. 

The path here struck into leads quickly into the higher regions 
of the subject ; we can follow it but a few steps, to find out its 
general direction. 

154. Art. 30 has already introduced a new kind of Cds. In 
the Eq. vjA^i + v^^^ + 1^3-^3 = of a RL. we may treat any 
two of the ratios of the three -^'s to each other as Cds. ; or, 
still better, we may treat the N's themselves as Cds. In this 
case the homogeneity of the Eq. in JVs shows that the apparent 
number of Cds., three, may at once be reduced to the real, ttvo, 
by dividing by any one. In fact, if ti, to, xg be the length of 
the sides of the A of reference, it has been shown that the 
triangular Cds. of any point are connected by the relation 

Hence, any two of such Cds. being known, the third is 
known. Again, any two determine a point. For all points 
distant Ni from Ni = lie on a RL. H to it ; so all points 
distant N^ from As = lie on a RL. H to it ; and these two 
RLs. meet in one, and only one, point. It is equally clear that 
any point determines two Cds., aud therewith the third Cd. 
Thus it is seen that points and triangular Cds. determine each 
other exactly as points and Cartesian Cds. 

If an}' two ratios of the A^'s be taken as Cds., then are these 
Cds. of 0th degree in the A's, and any combination of them will 
still be of 0th degree ; hence any Eq. between them will be 
homogeneous of 0th degree in A's, and on multiplication by the 
least common multiple of the denominators will remain homoge- 



188 



CO-OKDINATE GEOMETEY. 



neous of some degree. Hence Eqs. in triangular Cds. are 
homogeneous. 

155. Let us note more closely the Eq. of a RL. in triangu- 
lar Cds. : 

1/1^1 + ^2^2 + ^'3^3=0. (L) 

It is seen that the y's enter the Eq. exactly as the JSf's do. 
The significance of this fact is now to be developed. 

Holding the v's fixed and letting the JSf's vary, we get various 
points on the same RL. ; the RL. is fixed by fixing the v's ; any 

point on it is then fixed by 
fixing the JV's. If, now, we 
hold the JV's fixed and let the 
T/'s vary, we shall clearly get 
various RLs. thron<2,h the 
same point ; the point is 
fixed b}' fixing the JSf's ; any 
RL. through it is then fixed 
by fixing the v's. Thus it 
seems that the v's determine 
a RL. precisely as the ^'s 
determine a point. 

The JSf's determine a point 
as being its distances from 
three fixed RLs. forming a 
A ; how do the v's determine a RL. ? This question is easily 
answered thus : Take as the fixed point the vertex of the 
referee A counter to the side ]^i=0, i.e., the point 
(iV2 = 0, iVg^O); then Eq. (L) reduces to T/ii\^i = 0. 
Now for this point JS^i is 7iot = 0, hence vi= ; i.e., when 
all the RLs. go through the point (JV2 — 0, JV3= 0), then 
for all such RLs. ?'i = 0. Also, we know from Art. 29, 
that if I'l = 0, then all the RLs. given by Eq. (L) go through 
the point (^2= ^1 ^3=^)- Hence vi must be a factor of, or 
proportional to, the distance of the RLs. from the point 
(iV2 = 0, iV3 = 0), since when vi vanishes, and only then, the 




RIGHT LINES OF AN ENVELOPE. 189 

RLs. go through that point. Hence we may say vi = is 
the Eq. of that point, meaning all RLs. through it are distant 
from it 0, just as we say Ny = is the-Eq. of a RL., mean- 
ing all points on it are distant from it 0. Likewise vg = 0, 
V3= are the Eqs. of the other vertices of the A, counter to 
the sides whose Eqs. are ^2= 0, -^3 = 0. 

As we say the point (Ng, N^ , meaning the junction-point of 
the RLs. ^2=0, ^3 = 0, so we say the RL. (y,, I'a), mean- 
ing the junction-line of the points vg = 0, I's = 0. 

Had we not assumed the Eq. of the RLs. in the normal form, 
but taken the general form Aii^^ -|- Agig + >V3Z/3= 0, the 
reasoning would have remained unchanged. Hence, 

Triangular Cds. of a point are (fixed multiples of) its dis- 
tances from the three sides of a A. 

Triangular Cds. of a ML. are (fixed multiples of) its distances 
from the three vertices of a A. 

The above interpretation of the vs is indeed clear from Art. 
30. Accordingly, Eq. (L) may be interpreted either as the Eq. 
of a ML., the v's being arbitrary and the A'^'s variable, or as 
the Eq. of a point, the A^'s being arbitrary and the v's variable. 

The RL. fixed by the y's chosen at will holds on it every point 
whose Cds. (Ai, N^, A3) satisfy the Eq. (L). 

The point fixed b}' the A^'s chosen at will holds through it 
every RL. whose Cds. (vi, v^^ vo) satisfy the Eq. (L) . 

The RL. is the locus of the point (A^i, N^, A3) ; the point is 
the envelope of the RL. (vj, 1/2, T3) . 

Thus far have been used two sets of symbols A^'s and v's as 
Cds. resp. of a ijoint and of a RL. ; but that was plainly 
unnecessary, since the two sets enter Eq. (L) in the same way. 
We might as well say, regarding either set not as Cds., but as 
arbitraries, the Eq. (L) is the Eq. of a RL. or of a point, 
according as the other set be taken as the triangular Cds. of a 
point or of a RL. 

156. The picture of an Eq. of higher degree between point- 
Cds. is a locus, the Cds. of each of whose points satisfies the 
Eq. 



190 CO-OEDINATE GEOMETRY. 

The picture of an Eq. of higher degree between Une-Cds. is 
called an Envelope, the Cds. of each of whose RLs. satisfies 
the Eq. 

What is meant bj^ a point of a curve is well-known ; what is 
meant b}' a RL. of a curve is to be found out. 

B}' heaping together ever thicker and thicker points of a 
curve, the curve itself is not made but shadowed forth ; so by 
heaping together RLs. of an envelope, the envelope is shad- 
owed forth. Let 1, 2, 3 be any three such RLs. whose Cds. 

fulfil the Eq. of the envelope. 
Hold 1 fixed, and let 2 turn 
toward 1, its Cds. all the while 
fulfilling the Eq. ; its section- 
point with 1 will move along 1, 
and will be definite for every 
position of 2 ; as 2 falls upon 1, 
becomes coincident ivith 1, it is 
named consecutive with 1, and 
its section-point with 1 is named 
a point of the envelope. The 
student will see at once that this 
reasoning is quite parallel to that in Art. 64. If now we let 3 
turn toward 1, its Cds. all the while fulfilling the Eq., its section 
with 2, as it falls on 2, will be another point of the envelope ; 
these two section-points (1,2) and (2, 3) are plainly consecu- 
tive points of the envelope ; hence 2, which goes through both, 
is a tangent to the envelope. Hence a RL. of an envelope is a 
tangent to it; and on this account the Cds. of the RL. are com- 
monly called the tangenticd Cds. of the envelope or curve. 

157. It is now easy to see the meaning of the degree of a tan- 
gential Eq. The j)oint-Eq. (i.e., the Eq. in point-Ods.) of a RL. 
being of first degree, for a point-Eq. of a curve to be of 72th 
degree meant there were n points common to the curve and a 
RL. ; so, the tangential Eq. of a point being of first degree, for 
a tangential Eq. of an envelope to be of nth. degree means there 




TANGENTIAL EQUATION OF THE CONIC. 191 

are n RLs. common to the envelope and a point ; i.e., through 
a point may be drawn n tangents to the envelope. Such an 
envelope or curve is said to be of nth class. 

Again, the condition that a HL. should touch a curve was 
that on combining the two point-Eqs., two sets of point-Cds. 
should fall out equal, two common points be consecutive ; so, the 
condition that a point shall be on an envelope is that, on com- 
bining the Eqs., two sets of line-Ccls. shall fall out equal, two 
common RLs. be consecutive. 

This furnishes a general method of interchanging point-Eqs. 
and tangential Eqs. Combine the point-Eq. (L) of a RL. with 
the point-Eq. of any curve ; express the condition that two roots 
of the resulting Eq. he equal; the Eq-. which states this condi- 
tion states that the RL. whose Eq. is (L) touches the curve ; 
hence it is the tangential Eq. sought. In it the parameters 
(or v's) in (L) are tangential Cds. Likewise, combine the tan- 
gential Eq. (L) of a point with the tangential Eq. of a curve ; 
express the condition that tivo roots of the resulting Eq. he equal; 
the Eq. which states this condition states that the point whose 
Eq. is (L) is on the curve ; hence it is the point-Eq. sought. 
In it the parameters (or JV's) in (L) are point-Cds. 

Note that the analytic work in both cases is the same. 

158. For the curve of second degree a more elegant method 
is this : 

Common Cartesian Eqs. may be made homogeneous by 
replacing x and y by x:z and y:z, which plainly we may do, 
and then multiplying by z. The Eq. of a RL. becomes 

Ix + my -{-nz = ; 

the Eq. of a conic and the tangent to it at (cCi, i/i, z^) become 

kx^ -f- 2 hxy -{-jy^ + 2gxz -f 2fyz -\-cz^ = 0^ 

(kxi -f %! -f ^%) X -f {hx, -j-jy, -\-fz:) y 



192 CO-ORDINATE GEOMETRY. 

If the RL. be a tangent, then the first and third of these Eqs. 
can differ only by a constant factor /x ; hence 

kx^ -\- hfji -\- gzi — [xl == , 

^^^1 +jyi + A - /^^^ = 0, 

Also, as the point (xx, ^/i, ^i) is on the tangent, the Eq. holds : 

Ixi-{-myi-{-nZi = 0. 

The condition that these four Eqs. between Xi, 2/1, Zi consist, 

is 

= 0, 



k 


h 


9 


I 


h 


J 


f 


m 


9 


f 


c 


n 


I 


m 


n 






or KP -f 2i?Zm + M -{-2Gln + 2Fmn + Cn^ = 0. 

Such, then, is the tangential Eq. of the curve of second degree. 
The tangential Cds. are ?, ?7i, 7i, which are to be interpreted like 
the v's, while the capitals are, as always, the co-factors of the 
like small letters in the discriminant A. The relation of the 
above determinant to A is to be carefully noted. 

From this tangential Eq. the original point-Eq. is now to be 
got exactly as this tangential Eq. was got from the original 
point-Eq. : I, m, n will change back into a?, y., 2, while the capi- 
tals will change into their own co-factors in the discriminant 
I KJC I of this Eq. The result can differ from the original Eq. 
by a constant factor only, by which we may divide. This factor 
itself is readily found thus : 

Call the co-factors of K^ H^ etc., k\ h\ etc. ; then 

I KJ C\ = \kjc\\ and | k^ f c' | = | KJ C | ^ ; 
.-. |^"'/c'| =\kjc\\ 
But I AZj, Aj, Ac I = I A;J c I* ; 

whence Zj'= AZc, etc., 



beiaxchon's theorem. 193 

as the student may readily verify. Hence the constant factor 
is the discriminant A. When, and only when, A = l will 
7t' = k, or the deduced be the same as the original Eq. ; A can 
always be made = 1 by dividing the original Eq. by V A. 

159. Since 7c, h, etc., are at will, their co-factors K, H, etc., 
are at will ; hence the tangential Eq. of Art. 158 is the most gen- 
eral tangential Eq. of second degree ; since it represents a conic, 
we conclude that the general tangential Eq. o/ second degree repre- 
sents a conic. Again, since K, H, etc., are at will, so are Z:', 7i', 
etc. ; hence the point-Eq. deduced from the tangential is the 
general point-Eq. of second degree ; hence the general tangen- 
tial Eq. represents ever}" conic ; i.e., all curves of second degree 
are all curves of second class. In general, degree and class of a 
curve are not of the same number. 

If n points be taken on a conic and numbered consecutively 
from 1 to n, and each pair of consecutives be joined by a RL., 
the nth being joined to the first, the points will form the vertices, 
and the junction-lines the sides, of an inscribed n-side. 

If n tangents be taken on a conic and numbered consecutively 
from 1 to n, and each pair of consecutives be joined b}' a point, 
the 7?th being joined to the first, the tangents will form the sides, 
and the junction-points the vertices, of a circumscribed ?i-side. 

In this way we might now proceed to double by re-interpreta- 
tion the whole bod}" of doctrine gone before, by making proper 
chansres in words throughout. But such detailed treatment 
would not be in place here. One special case of great import- 
ance may serve to illustrate. 

160. If i?i = 0, ^2 = 0, i?3=0, i74=0, i?5=0, 

JIq = are the Eqs. in homogeneous point-Cds. of the sides of 
a 6-side inscribed in a conic, they are also the Eqs. in homoge- 
neous line-Cds. of the vertices of a 6-side circumscribed about a 
conic : the Eqs. of the two conies will be the same in form, but 
one will be in point-Cds., the other in line-Cds. By Pascal's 
Theorem the junction-points of H^ — O and H^ = 0, jSg = 



194 



CO-ORDINATE GEOMETRY. 



and 115 = 0, ^3 = and IIq = lie on a RL. ; the Eq. 
which says this, interpreted in point-Cds., saj's, when inter- 
preted in line-Cds., that the junction-lines of IIi=0 and 
i?4=0, H2 = and ^5 = 0, 11^ = and i?6 = 0, meet 
in a point. Thus is found Brianchon's Theorem : 

The three diagonals through the counter-vertices of a 6-side 
circumscribed about a conic meet in a point. 

This correlate to Pascal's Theorem was first proved, though 
not as above, b}' Brianchon, a pupil of the Polytechnic School 
at Paris (1827). As tliere are sixty Pascal lines (Pascals), so 
there are sixty Brianchon points (Brianchons) . 

By Pascal's Theorem we can find a sixth point of a conic, 

knowing 5 ; by Brian- 
chon's we can find a 
sixth tangent, know- 
ing 5. For be 1, 2, 3, 
4 the junction-points 
of the pairs of consec- 
utive tangents, taken 
in order ; take on the 
fifth tangent any point, 
as vertex 5 ; draw 1 4 
and 2 5 ; through their 
section and 3 draw a 
RL. ; it will cut the first side at vertex 6 ; then is 5 6 the sixth 
tangent. 

By Pascal's Theorem we could find the tangent at any vertex; 

by Brianchon's, we can find 
the tangent-point on any tan- 
gent. For, suppose the tan- 
gent or RL. 5 6 to fall 
together with 6 1 upon 5 1 ; 
then 5 1 touches at 6. Draw 
1 4 and 2 5 ; through their 
section and 3 draw a RL., the third diagonal; it will cut 5 1 
at 6. 





EECIPKOCAL CXIKVES. 195 

161. Line-Cds. are of special use in dealing with loci of 
poles and envelopes of polars. If the pole (as to any referee) 
move on any curve L, its polar will turn around some curve E. 
The junction-line of two points of L is the polar of the junction- 
point of the polars of those points ; if these points be consecu- 
tive, their junction-line is tangent to L ; then their polars are 
consecutive, and the junction-point of these polars is the point at 
which they, fallen together, touch JE. Hence the iiolars of all 
points ofE are tangent to L; i.e., as thepo/e traces L i\iQ polar 
envelops E, and as the poZe traces E the polar envelops L. 
Hence the relation of L and E holds when the terms are 
exchanged; i.e., it is a ??^^^fwaZ relation. L and E are called 
7'eciprocal curves as to the referee ; this may be any conic, most 
simply a circle. 

We ma}' now prove Brianchon's Theorem from Pascal's thus : 
The poles of the six sides of the hexagon inscribed in L are the 
vertices of a hexagon circumscribed about E. The junction- 
points of pairs of counter-sides of the inscribed hexagon are 
poles of the diagonals (junction-lines) of counter vertices of 
the circumscribed; since the poles lie on a RL., the polars go 
through a point. Since L is any conic, so is E. 

Of course, it is just as easy to prove Pascal's Theorem from 
Brianchon's ; it is done b}' exchanging clauses in the sentence, 
" since the poles, etc." Neither theorem is logically first. 

These methods of double interpretation and of exchanging 
the notions of pole and polar have received the names of Prin- 
ciples of Duality resp. Reciprocity. They are in last analysis 
one, and their possibility is given in the fact that tlie plane, 
whether viewed as full of points or full of ULs., is doubly 
extended: there are as many points as RLs. in the plane; a 
point for every RL., a RL. for every point. The referee sets 
these points and RLs. in relation to each other. 

In conclusion, it is to note in regard to reciprocal curves 
that, if a RL. cuts L in n points, through its pole go n RLs., 
polars of those points, all tangent to E ; hence the degree 
(resp. class) of either of two reciprocal curves is of the same 



196 CO-ORDINATE GEOMETRY. 

number as the class (resp. degree) of the othei^ Hence the 
point- (resp. tangential) Eq. of either will be of the same 
degree as the tangential (resp. point-) Eq. of the other. Hence 
the reciprocals of conies are conies: if the pole traces a conic, 
the polar envelops a conic ; and conversely. 

Note on Points and Right Lines at Infinity. 

In view of the extensive and important use made of the notions of 
point and RL. at oo, it may be well to ground these notions more thor- 
oughly than could be done in the body of the book without breaking quite 
the thread of thought. 

All reasoning is in first intention not about things, but about notions or 
concepts. It is a familiar fact of every-day life that the same thing may 
be conceived variously, and that the conclusions that hold about it may 
vary accordingly. Important illustrations have already met us. Two 
coincident points are in themselves one and the same point; it is only the 
mind that thinks the point now as on this curve, now as on that. So con- 
secutive points are in themselves one and the same ; it is only in thought 
that they are held apart. 

The obverse of this fact is the less familiar one that things in them- 
selves different may be, indeed, must be, thought as the same if there be 
no mark to distinguish them in thought. The conclusions that hold about 
them will be the same. Such things are all points on a RL. whose dis- 
tances from any given point of that RL. are unassignably great. How- 
ever apart they may be in themselves, they cannot he held apart in thought. 
Hence all such points are, for thought, one point, and we speak with strict- 
est accuracy of the one point at co, the one point, not of the RL. out of 
thought, but of the RL. in thought. 

Such, again, are all points at co ori parallel RLs. If y = ^ and 
y — h be two such RLs., then indeed the oo points of the two would seem 
to have this mark of distinction, that the y of the one is while the y of 
the other is h. But this difference, while it distinguishes them in their 
outer being, does not yet distinguish them in thought. For it imparts no 
property to the one that does not belong to the other. By the side of the 
00 value of x, the finite value of y loses all distinguishing power. This is 
clearly seen on drawing a RL., say through the origin, towards the oo 
point of the RL. y=b; the RL. is clearly none other than the X-axis, 
y = 0, for any other RL. will not extend toward the oo point of y = b, 
but toward some definite finite point of it. Hence we say all parallel RLs. 
meet at oo, meaning that the marks of distinction in our notions of the 



EXAMPLES. 197 

points of the RLs. vanish utterly as the points are thought retiring on the 
RLs. without limit, leaving the notions of all the points undistinguished, 
one and the same. That parallel RLs. meet at co is, then, no merely con- 
venient form of speech, but states a/ac^, not of things but of thoughts. 

Like reasoning applies to the RL. at oo . It might indeed be said that 
there are many RLs. at oo , for any RL. might be thought pushed to co 
while kept in a given direction ; but all such RLs. lose all distinction of 
direction in thought, yea, since each must go through the co point of each 
axis, they fall together in thought completely. Hence the phrase the RL. 
at CO correctly expresses our thought of them. 

Let the student beware of confounding the notion or concept of a thing, 
which is given in its definition and is the subject-matter of thought about 
it, with its mental image. Of points and RLs. at co there are no such 
images at all. 



EXAMPLES.* 

Centre and Diameters. 
1. Find the centre and the pencil of diameters of 

5x2_^ 12a:y-3/ + 8x-10;/-7=0. 



5 6 4 

6 -3 -5 

4 -5 -7 



= 5 (_4)-6(-22) +4(-18) =40, 



(7=-51, G^-22, i^ = -18; 

(90 "I Q\ 
^ - 1 : the pencil of diameters is 
51 51 y' ^ 

5x-f6?/ + 4+A(6:r — 3y — 5)=:0; 
the central Eq. is 

5x2+12a:?/-3?/2-^ = 0; 
51 

the curve is an H. 

2. Discuss in like manner these Eqs. : 

3x2_8:cj/+7?/2-4a:+6 3/=13; 
4 a:2 — 6 a: ?/ + 9 ?/ 2 + 5 a: + 3 ?/ = 1 ; 

7 a:2 _ 10 T?/ + 3 / — 8 X — 12 ?/ = 2 ; 

* It has been thought best not to interrupt the development of the sub- 
ject, but to put all the exercises at the close. The teacher may introduce 
them as he deems fit ; they are arranged in the order of the foregoing text. 
For very many the author has to thank Hockheim's admirable collection. 



198 CO-OHDINATE GEOMETRY. 

2ar2- 5x^-3/ + 9^-13?/ + 10=0; 
3 2/2-2a:^ + ?/2 + 2a; + 2^ + 5 = 0; 
13a:2+14;r// + 6/+ 14a: + 10?/ + 5 = 0. 

3. Turn the axes so as to make the term in xy vanish. 
Be kx^ + 2 hxy +jy'^ + c = the central Eq. ; put 
X = x^ cos a — y' sin a, y = x' sin a + ?/' cos a ; 



on substitution — 2^ sin acos a+ 2j sinacosa + 2 A (coso — sina ) must 

2 A 
= ; hence tan 2 a = , and the Eq. takes the form 



[k J^j _ V{k -jy + 4 h^y^ + [Z: +y +V{k -jY + 4 A2] ?/''^ + 2 c = 0. 
Illustrations: 10a:2-6:r3/ + 7/ = 30 ; 9a;2 + 16:r^ -20 j/2 = 60. 

4. Turn the axes till the X-axis is II to the axis of the P 
kx'^ + 2 hxy ■^jy'^ + 2gx -^^fy + c = {}. 

Here C= kj —h? = 0, or A = Vkj; the axis of P is II to 

Ik k h 
hence tan a= —\- = = ; the reduced Eq. is 



J^±j 



,/^ + 2 I ^f-^9\^, + 2 f ^^+ ^^ \' + c = 0. 



Illustration : 9 j:^ _ 6 ^^ ^ ,^2 _j_ 4 3. ^ 3 ^ ^ 10 = 0. 

5. A diameter of 1- ^i- = 1 is y = sx; what is the conjugate '? 

6. Find the chord of — + ^=1 through (1,2) and halved by 
2y = Sx. 

7. A diameter of ^ = 1 is y = sx; find the conjugate. 

8. Eind the chord of ^ +t- = l halved by (4, 2). 

36 9 J' V . ; 

The diameter through (4, 2) is 2 y = x ; the conjugate is 2y ■{■ x = 0; 
hence the chord is 2(^ — 2) + (a: ~4) =0, or 2y + x = 8. 

2 2 

9. Eind the chord of — — ^- = 1 halved by (5, 3). 

4 49 J' \ ' I 

10. In ^ = 1 find the Y between 3 3/ = :r and its conjugate. 



EXAMPLES. 199 

11. In 25 x^ — 16 3/2 = 400 find the conjugate diameters sloped 45° to 
each other. 

12. Find where the diameter through (2, 3) and its conjugate cut 

4 x^ + 12 2/2 = 48. 

13. Find lengths of the diameter 4c2/= 5x and its conjugate in 
49^ + 9x2 = 441. 

14. Find the slope of the diameter through (a:^, y^) to its conjugate. 
The direction-coefficients of the two diameters are '^^- and t — •-^-', 

hence tan d> — - — % ^ = + , since a'^y^^ ± b'^x-^^ = ± a'^b'^. 

o? 

15. In — + -^ =r 1 find the conjugate diameters sloped 120° to 

362 52 

each other. 

16. Find the length of the diameter conjugate to 2 a; =: 5 2/ in 

-4x'^ + 25/ = -100. 

17. Given a' = 1, 6' =10, (^=110°; find a and 6. 
Remember the relations a'2 4- 6'2 — ^2 _j. j2^ ^^/^^ gjjj ^ _ ^j_ 

18. Find the slopes to the axis major of the equi-con jugate diameters 
of 64 y2 4. 25 x^ = 1600. 

19. Prove that the diagonals of the parallelogram of tangents at the 
ends of conjugate diameters are themselves conjugate diameters. 

20. Form the axial Eq. of the conic : when a = 13, ae = 12 ; when 
a + 6 = 27, ae = 9; when the conic goes through (1,4), (—6,1); when 
it goes through {^lyyi), {^iyV^I'i when 6'^ = — 144, ae = 13 ; when 
e = 3; when a = 4 and the conic goes through (10, 25). 

21. Find the points of a centric conic for which x and y are equal. 
When and how can these points be constructed 1 

22. Find the axes of the conic whose vertical Eq. is y"^ = h^x — ff ar2, 

23. Find the vertical Eq. when & = 6 and the parameter = 5. 

24. An H goes through {x^, y^) and the parameter is q ; find the vertical 
Eq. and the length of the real axis. 

25. Express the vertical Eq. of H through e and b as known. 

26. Show from the central polar Eq. of a conic that the sum of the 
squared reciprocals of two ± diameters is constant. 



200 CO-ORDINATE GEOMETRY. 



27. Find where — ± -^ = 1 is cut by 2/ = sx + c. 

a' b'^ 

28. Interpret geometrically d^s'^ ± b'^ — c^ = 0. 



> 



Since s — tan 9, we may write a^ = ah^ cos + c^ cos 6 ; hence the 

RL. cuts the £ in real, coincident, or imaginary points, according as the 
foot of the focal ± on it lies within, upon, or without the major circle ; 
vice versa for //. 

29. When does kx^ + 2 hxy +jy'^ + 2gx ■{- 2 ft/ + 0=0 touch either 
or both axes ? 

30. Pind the Eq. of the tangents from the origin to 

kx^ + 2 hxij -\-jy'^ + 2 ^x + 2/^ + c = 0. 

31. Eind the Eq. of the RL, halving each of the positive half-axes of a 
centric conic ; where does it cut the conic 1 

32. Two vertices of an equilateral hexagon inscribed in the £ 
25 y"^ + 9x'^ = 225 are at the ends of the axis minor ; where are the others 1 

33. Inscribe in a centric conic a rectangle of area 2 ab. 

Tangents and Normals. 

34. Tangent and normal form with the X-axis an isosceles A; find its 
vertex, the point of tangence. 

35. At corresponding points of an £ and its major circle are drawn 
tangents ; prove that the subtangents are equal. Hence frame a rule for 
drawing a tangent at any point of an £. 

36. Find the Eqs. of tangents to 5?/^ -f- Sa:^ = 15 il to 3?/ — 4a;— 1 = 0. 

37. Find Eqs. of tangents to 36 / -f 25 x^ = 900 sloped 30Mo the 
X-axis. 

38. Find the Eq. of the tangent to 9 ^^ -f x^ = 9 when the X-tangent 
is 5. 

39. Find the point of tangence whose abscissa equals the subtangent. 

40. Form the Eq. of a circle whose diameter is the tangent-intercept 
between the vertical tangents. Where does it cut the X-axis ? 

41. Form the Eq. of a circle whose centre is on the axis minor, and 
which has the Y-tangent as a chord. Where does it cut the other axis ? 

42. Find the Y between the tangents at [x^, y-^) and [x^, y^) and the Eq. 
of the diameter through their intersection. 



EXAMPLES. 201 

43. Find the ratio of the rectangles of the ordinates resp, abscissas of 
the points of touch of two ± tangents. 

44. Find tlie common tangents to a centric conic and its mid-circle 
[x^ -\- y'^ T= ah) , also the Y under which the curves cut. 

45. Find the Y between the tangents from (a:^, y^) to a centric conic. 

Foci and Directrices. 

46. From the foot of a directrix (on the axis) is drawn a tangent to a 
conic ; from any point of the tangent is dropped a ± on the axis ; from 
where the ± cuts the curve is drawn a ray to the focus ; find the ratio of 
the ± to the focal ray. 

47. Find the Cds. of the pole of Ix -\- my + n — Q as to a conic. 

48. Draw a tangent to a conic at a given point, a focus and its direc- 
trix being known. 

49. The diameter through P (^p^i) ^uts a directrix at D', find the Y 
between the polar of P and the focal ray of D. 

50. Given an Y, the counter-side, and the sum or difference of the 
other sides of a A; find the sides and angles. 

51. Find the sum of two focal chords II to two conjugate diameters. 

52. Find the rectangle of the segments of a focal chord. 

53. Find the harmonic mean of the segments of a focal chord. 

54. Find the harmonic mean of two ± focal chords. 

Asymptotes. 

55. Find the Y between the asymptotes of 4 o:^ — 5 ?/^ = 100. 

56. How long is the focal ± on an asymptote 1 How long is the 
asymptotic intercept between the two ±s ? 

57. Find the Eq. of the tangent at {x^, y^) to 4 xy — a"^ + P. 

58. When does y = sx -\- c touch 4 xj/ = a^ -}- 6^ ? 

59. Find the Eq. of the RL. through (.r^ y^), (x,2, y^) on 4 xy = a? -\- b^. 

60. The asymptotic intercept between two chords joining two fixed 
points of an /^ to a variable point of the H is constant. 

61. Find the asymptotic Cds. of the pole of the chord through {x^^, y^), 



202 CO-OEDINATE GEOMETRY. 

62. Find the asymptotic Eqs. of the dh'ectrices. 

63. The asymptotic distance of a point of an H from a directrix equals 
the focal distance of the point. 

64. The focal ray of a point of a conic, the ± through the focus on the 
ray, and the polar of the point go through a point. 



The Parabola. 

65. The vertex of a P is (a, h), t\\^ parameter is q, the axis is II to the 
X-axis ; what is the Eq. of the P 1 

66. The axis of a /* is y = 6, the x of the vertex is 2, and the curve 
goes through (7, — 8) ; find the Eq. 

67. Eor what point of ?/^ = 4 qx is ij «-times x ? 

68. What is the Eq. of y^ = 10 x when a>=45^, the a.\is being a 
diameter and the tangent through its end ? 

69. Eind the rectangle of the ordinates of the ends of a focal chord. 

70. Where does y = 6x + c cut y"^ = 4:qx 1 Interpret q = sc. 

71. Eind the side and height of an equilateral A inscribed in a P. 

72. Eind the Eq. of the chord through {x^, y^) that is halved by {x^, y^). 

73. Eind the Eq. of the P whose axis is II to the X-axis, wliose param- 
eter is 3, which cuts the X-axis at (12, 0), and touches the Y-axis. 

74. Find the Eqs. of the tangents from the origin to 

[y — h)'^ = 4 (/ [x — a). 

75. The vertex of a P is at (a, 0), its axis falls on the X-axis, and it 
is touched by Ix -\- my -\- n — ; find its Eq. 

76. Find the tangent-lengths and normal-lengths in P. 

, 77. Find the Cds. of the point of touch of a tangent sloped (p to tba 
axis. 

78. When is the normal-length equal to the difference of subtangent 
and subnormal 1 When is the rectangle of the tangent- and normal- 
lengths equal to the square of the ordinates ? 

79. Find the ratio of the tracts drawn from any point of a tangent to 
the focus, and the foot of the ± from the point of touch on the directrix. 

80. Find the Eq. of a tangent to y'^ = 4 gx sloped ^ to ?/ =r sx -f c. 



EXAMPLES. 203 

81. Find the rectangle of the subtangents of two ± tangents. 

82. Show that tangents at the ends of a focal chord are ±. 

83. Eind the Eg. of the focal ray of the intersection of tangents at 
P P 

84. An isosceles A is circumscribed about a P ; show that the vertex, 
the point of touch of the base, and the focus lie on a RL. 

85. An equilateral A is circumscribed about a P ; the transversals 
through the vertices and the points of touch of the counter-sides go through 
the focus. 



86. Under what Ys do x^ + ^^ _ ^2 ^^^ x — q -\- y"^ — p- cut 

87. Eind a P whose axis falls on the + X-axis, and which touches 

9 

y^ ■\- [x — aY = r"^ enclosing it, e.g., y^ -f a; — 13" = 25. 

88. Eind the common tangents to o-^ + ?/2 = r"^ and ?/^ = 4 qx. 

89. Eind the common tangents to the co-axial P's ^^ = 4 q-^x, 
if=4:q^[x-a). 

90. Eind the Y between the tangents through {x-^, y-^) to y'^= 4:qx. 

91. What is the pole of Ix -{- my -\- n — as to y^ = 4:qx1 

92. P is on the directrix of P; F is focus; the polar of P meets the 
curve at /, /'; PPis geometric mean of IF and FF. 

93. The diameter of a /' through P cuts the directrix at D; show that 
FD is ± to the polar of P. 

94. The axial intercepts of the polars of two points as to P equals the 
axial intercept of _Ls from them on the axis. 

95. The Eq. of a pencil of II chords of a Z' is y = sx -]- b ; what is the 
Eq. of the conjugate diameter ? 

96. If a chord cuts off equal segments from two diameters of a P, the 
diameters cut off two equal segments from the chord. 



General Focal Properties. 

97. The focal ± on a tangent meets the central ray to the point of 
touch on the directrix. 

98. The focal ray to a point of a conic equals the ordinate of the point, 
prolonged to the tangent at an end of the focal ordinate. 



204 CO-OEDINATE GEOMETEYo 

99. The focal polar Eq. of a chord, the slopes of the rays to whose 
ends are 9-^ + 6^, 6i — 62, is 2q: p = — e cos 6 + sec 62 cos (6— d^). 

100. Hence, show that the tangent at {p^, 6^) is 

2q: p — — e cos + cos {6 — 6^). 

The Eq. of a RL. through {p', 6') and (p", 6") is 



pp' sin e — 6'+ p'p" sin 0' - Q" + p"p sine" —6 = 0. 
Put 0^+ ^2 resp. 6^ — O^ for ^' resp. Q" , and for p' resp. p" put 

^ • resp. , 

1 — e cos 6' 1 — e cos 6" 

since {p',&), {p",Q") are on the curve, clear of fractions; so we get, on 
collecting, 



p sin (0—01 + 02) (1—e cos 01 — 0.) +2gsin2 02 



+ psin(0i— 02 — 0) (1 — ecos 0^+ 02) = 0, 
2 p sin (—02) cos (0— 0^) + 2gsin202 



— ep{sin (0—01+ 02) COS01 — 02— sin(0 — 01— 02) cos0i + 02} = O. 
Applying the addition theorem of the sine to the bracket, we get 



2 p sin (— 02) cos — 01 + 2 ^^ sin 2 02 + e p sin 2 02 = 0, 

whence, on transposition and division by sin 2 02 = 2 sin 02 . cos 02, there 
results the Eq. of Ex. 99. In Ex. 100, 02 = 0. 




101. If be fixed, PP' any chord through it, then tan JPFO. tan JP'FO 
is constant. 

By 99, the Eq. of PP' is 



25':/j= — e cos 0+ secj8 + jS'cos (0 — 0+ /8 -\- ^'). 



EXAMPLES. 



205 



Hence, 2^ : p^ = — e cos a + 2y8 + secyS + j8' cos (^8 — j8') for the point 
0{p^,a + 2B). 

Or, cos )3 — ;8' : cos 13 + ^' is constant ; 



or, 
or. 



{cos ;8 — ;3' — COS j8 + j8'} : {cos ;3 — j8' + cos j8 + fi'} is constant ; 

^^ - = tan /3 . tan /3' is constant. 

cos/3 cos/6' 



102. Normals at the ends of a focal chord meet on the 11 to the axis 
major through the mid-point of the chord. 

For they meet in the centre of the circle through the ends of the chord 
and the other focus ; the proof is now readily completed. 

Hence, find the locus of the intersection of such normals. 



103. Find the locus of the intersection of -L normals to a P. 
The magic Eq. of the normal is y — sx -{-2qs — qs^ = 0. The 



normal 



is q -\- 2 qs^ 



xs^ 


-ys^ = 




These 


Eqs. consist when, 


and 










y 


x — 2q 





-q 





q 


X 


-2q 





-q 





y 


X — 2q 







-1 
















1 





2q — x 


y 





<1 







2q — x 


y 





? 





2 


q — X 


y 









0. 



Multiply the first row by y, and the third by q ; add to the first q times 
the fourth ; take from the third y times the sixth ; there results 





y 

qx — 2 q^ 
9 



q^ + y"^ 

X — 2q 

xy-2qy 





xy — ^qy 


2q — X 



2q'^ — qx 

-q 



y 



= 0. 



Multiply the last row by q, add to it y times the second, then add to 



first row 2 q — X times the second ; there results 



= 0. 



y {2 q — x) q"^ -\- y'^ — {2 q — x)"^ y{x — 2q) 

q{x-2q) y{x-2q) - {q^ -^ y^) 

y2 ^ q2 y^x — 2q) q {^ q — ^) 

Take the second row from the third, set out the factor y'^ — qx -\- Sq^ ; 
this factor equated to satisfies the Eq. of condition, making the determi- 
nant 0, and is the locus sought : a P one-fourth as large as the original, its 
vertex where normals at the ends of the focal chord meet. This result 
may be got more simply otherwise, but the above illustrates the general 
method and the use of determinants. 



206 CO-OEDINATB GEOMETRY. 



Eccentric Angle. 

104. The lengths of two conjugate half -diameters, a' and 6', when e is 
the eccentric Y of a', are 

105. Find the length of a chord of an £ in terms of eccentric Ys. 

\2 ~ oP- (cos ej — cos 62)^ + ^^ (sin e^ — sin Cg)^, 



or 12^=4sin^i ^ i a2 sin ^^^t_!2 + 52 ^os ^i-+i2 I. 

2 ( 2 2 ) 

The bracket is the squared half -diameter II to the chord; call it D^\ 
1 2 =: 2 2)3 sin ^^ ~ ^^ . Or, much more neatly, thus : 

The corresponding chord of the major circle is 

F2>= 2 D ' sin £i^:^i2. 
^ 2 

In projection II chords are changed in the same ratio; the II diameter 2a' 
or 2 D' changes to 2 D^, hence 

22)3/ sin ^1-=^-^ to 21)3 sin ^ 



2 " 2 

106. rind the area A of the A whose vertices are (ej, (eg), (63). 
The sides of the corresponding A' are 2 a multiplied by 

sin ^^ ~ ^ 2 sin ^2 ~ 3 resp. sin - ^ ~ ^ : 
2 2 ^ 2 

the double area of any triangle is the product of the sides divided by the 
diameter of the circumscribed circle ; 

.-. 2 A'=: 4 aa sin ^^ ~ ^^ . sin ^^ ~ ^^ . gjij^ 



2 
hence A = 2 a6 sin ' ^^ ~ ^''- - sin ^2 ~ ^3 . gj^ fs — fi 



Clearly, also, 



2 A'= aa {sin e^ — 63 + sin ^2 ~ ^3 + sin ^3 — ^j} ; 



hence sin e^ — eg + sin €2 — €3 + sin eg — e^ 



€0 — e. 



= 4 sin -^ 2 • sin -2 3 . sin -^ 



2 2 2 

If r be the radius of the circle about the A (e^, eg, eg), then 



EXAMPLES. 207 



2Ar=12.2 3.31:2r, or r=12.23.31:4A, 

ab 

A focal chord c^ is a third proportional to the axis major and the II 
diameter ; i.e.,- 2a : 2 Z)i = 2 Z>i : c^ ; hence results 



165- a 

107. Find the area A of the A of tangents touching at (e^), (e^), (eg). 
The area A' of the corresponding A' is plainly 



Af 



= aa \ tan ^1 -^ + tan ^2 !? + tan ^2 li I 

( 2 2 2 ) 



.-. ^ = a6 { . . . }. 

By applying the determinant formula for the area of a A fixed by three 
RLs. we find 

A = a6tan ^3JZ-L2 . tan ^2-=-^ • tan ^^s^^i ; 
2 2 2 

hence the sum and ^Ae product of the :fAree tangents are equal, a relation hold- 
ing only when the sM?n of the Ys is a multiple of tt. 

108. Show that the area of the A formed by three normals is 

ZEI'' / tan ^-IZ1£2 . tan '-2-=^ • tan ^2JZii I 
4a6 I 2 2 2 / 



{sin ei + 63 + sin €3 + €3 + sin eg + e^}. 



Areas. 

109. The side of a rhomb inscribed in an £ is s, the linear eccentricity 
is the geometric mean of the half-axes ; find the area of the £. 

110. In the £ 25 y'^ + 9 a:^ = 225 find the area of a sector whose 
centric Y reckoned from the axis major is 60°. 

111. Find the ratio of the segments into which the parameter cuts 
an £. 

112. Find the ratio of the parts into which a concentric circle through 
the foci cuts an £, 

113. Find the ratio of the parts into which a confocal P, with vertex at 
the centre, cuts an £. 



208 CO-ORDIKATE GEOMETRY. 

114. Find the segment cut off from IP'x^ — a^j/^ = a^lr^ by x = d. 

115. Find the area bounded by IP-x^ — a^y^ — a^h'^ and the RLs. 
J/ = (^, y = - c. 

116. Fmd the area bounded by an ^, its major circle, and y — o,, 

117. An equilateral A of side s has for its altitude the axis, for its 
vertex the vertex of a P; find the segments cut off by its sides. 

118. The segment cut off by a focal chord of a P is one-third of the 
trapezoid of chord, directrix, and diameters. 

119. A focal ray is prolonged by twice itself ; through the point thus 
reached is drawn a diameter to the P', compare the triangular areas 
bounded by the ray, the curve, the diameter, and the axis. 

If the ray meet the curve at {x, y), the ratio is found to be — ^, 

which is 10 Avhen the ray is ± to the axis. ^ ' ^ 

120. Find the ratio of the parts into which y'^ — A.qx cuts 

y"^ =L \Q) qx — x'^ . 

121. Find the area of the figure bounded by y'^—^qx and 0:^= ^qy. 

122. Find the area between y'^—^qx and y'^ — ^q{x — q). 

123. Find the area of a focal sector of a P. 



Determination of the Conic. 

124. Determine elements of 3 z^ — 4 x^ + 5 ?/2 - 30 x — 16 j/ — 20 = 0. 
A = -2017, 0-11, the centre [G:C,F: C) is [ -, — ], the curve 

2017 
is an E, the central Eq. is Zx"^ —^xy + by"^ — 0, the Eq. of the 

pair of RLs. through the centre and the intersections of the E and a con- 
centric circle is 3 x^ — 4 x// -f 5 ?/^ — . " — ^t-i_ — Q • these RLs. fall 

J^ J 11 ,-2 

together in an axis when, and only when, 

or 2017^_g_2()17^g^^__j5 -^g^g 

112 ,.4 11^2 



EXAMPLES. 209 



2017 



whence = 4 ± ■\/6: whence the axes are 

11 r2 

(V5+ l)a:2 + 4xy+(V5-l)/=0 

and (Vo"-l)a:2_4:ry+ (V5+l)/ = 0, 

{V6 + 1}» V5~+ 1 
or ?/ = — i — -—I — i- a: = ^ — x, 

{Vo-ip 2 

{V6-1]J V5-1 
and ?/ = ^ — — ^ :r = x ; 

{V5 + ip 2 

while the half-axes are respectivelj" 

V4- V5. V20I7 and V^ + V5 . \/2017 
11 11 

125. Determine the elements of 

x^ - 5 xij + y'- + Sx -20 1J + 15^0, 

36 a:2 + 24 x?/ + 29 _y2 _ 72 + 126 J/ + 81 = 0, 

2 ^2 - 5 x// - 3 / + 9x - 13.?/ -f 10 = 0. 

126. Determine the elements of 9 x^ — 6 xi/ -{- y'^ + 4 x -{- 3 y + 10 = 0. 
A = — 169 : 4, C = 0, the curve is a P; the Eq. may be written 

(3 :r - ?/ + A)2 = 2 (3 A - 2) X - (3 + 2 A) ^ + a2 - 10, 
or L^ = L'; 

the ELs. L = and L' = are a diameter and a tangent at its end ; 
they are ±, and .'. are the axis and the vertical tangent when 

3 2(3a-2) -^^Q 20a-9 = 0, a = — • 

2A + 3 20 

The parameter is 

4f3.1-2Y+4f-l.l-§ 



^ . , 20 / V 20 2/ J 13VIO 

^ 3'^ + 1^ 100 

127. Determine the elements of 

25 0:2-120 xy+ 144/-2a:-29?/==l, 
9 0:2 - 12 xy + 4 ?/2 _ 24 a: + 16 ?/ - 9 = 0, 
4o:2 + 9 ?/2 _ 8 0: + 54 ?/ + 85 = 0. 

128. The linear eccentricity of Jcx"^ + 2 hxy + jy^ + 2 ^ro: + 2fy + c = 

is V-V(I^^^7FT4F.A: a. 



210 CO-OEDINATE GEOMETBY. 

129. Find the conic through (0,0), (2,2), (18,6), (32,8), (72, 12) by 
inspection. 

130. Find the conic through (3,7), (-2,-8), (11,31), (9,-2), (17,1). 

131. Findjhe conic through (1,2), (0, 1), (1 : 4, ( V3 + 5) :4), (^,2), 
(7:8, (15- V7): 8). 

132. Find the conic through (-8,0), (3, Vll:12), (4, --2:\/3), 
(1,-1:2), (6, V7T3). 

133. Find the P touching the axes at (4,0) and (0,3). 

We have K=0, J^O, C=0, 16^+8(7 + c = 0, 9j+6/+c = 0, 
whence (3 j:±4 ?/)2 — 24 (3x -f 4_y — 6) = 0; to the lower in the double 
sign corresponds a P, to the upper a double RL. through the points of 
touch. 

134. Find the conic touching both axes, the Y- one at (0, 4) and going 
through (16 : 3, - 20 : 3), (- 3, 10 + | Vl5). 

Constructions. 

135. Given the conic drawn, to determine its elements. 

Draw a pair of pairs of II chords, and through the mid-points a pair of 
diameters; these meet in the centre. If the conic be a P, draw two chords 
_L to the axial direction ; their halver is the axis ; from the foot of any 
ordinate lay off a subtangent double the abscissa, through its end and the 
end of the ordinate (on the curve) draw a RL. ; it is a tangent to the 
curve. Through the mid-point of the tangent-tract draw a ± to it ; it meets 
the axis at the focus ; the directrix is _L to the axis and is counter to the 
focus as to the vertex. 

If the conic be centric, draw on any diameter of it a half-circle, and 
from where this meets the conic draw to the ends of the diameter a pair 
of chords; they are supplemental and ±; hence, the diameters II to them 
are conjugate and ± ; i.e., are the axes. In the E, from an end of the axis 
minor draw a circle with the half-axis major as radius ; it cuts this axis in 
the foci. But, in the H, the ends of the (minor or) conjugate axis being 
imaginary, draw the vertical tangents (± to the real axis) ; draw the diam- 
eter of any two II chords, and at its end a tangent ( II to the chords) ; on 
the intercept of this tangent between the I! tangents as diameter, draw a 
circle; it cuts the real axis in the foci. A circle about the centre and 
through the foci meets the vertical tangents on the asjjmptotes. Combining 
the Eqs. of an asymptote and the major circle, we see they meet on the 
directrices ; but this construction of the latter is available only in case the 



EXA^IPLES. 211 

asymptotes are real, i.e., in H. In the £, draw a tangent at the end of a 
focal ordinate ; it cuts the axis major on the directrix corresponding to 
that focus. , 

136. To draw a tangent at a point to a conic. 

In P, lay off from the foot of the ordinate the subtangent double the 
abscissa ; thus is reached a second point of the tangent. In E, draw a tan- 
gent at the corresponding point to the major circle; it cuts the major axis 
in a second point of the tangent to £. In H, halve the inner Y of the 
focal rays to the point. The like construction of course holds for P and E. 

137. To draw a tangent from a point to a conic. 

On a focal tract to the point, as diameter, draw a circle ; it meets the 
major circle in two second points of the two tangents. In P the vertical 
tangent is the major circle. 

138. Given the foci and one point (or tangent) ; construct the conic. 
To give the focus at oo is the same as to give the direction of the axis 

of R 

139. Given a focus, an axial direction, a tangent and its point of touch. 
Use the counter-circle of the other focus. 

140. Given 2 a, a focus, a tangent and its point of touch (or an asymp- 
tote). 

141. Given a focus, two tangents (and their points of touch in E and H). 

142. Given a focus, and one diameter in length and position. 
Find other focus and the axes, and use the major circle. 

143. Given three tangents and a focus. 
Use major and counter circles. 

144. Given a focus, two tangents, and the axial directions. 

145. Given the centre, a focus, and a tangent ; find where tangents 
from a point P touch the conic. 

On the focal tract FP as diameter draw a circle ; through its intersec- 
tions with the major circle draw RLs. from P ; they are the tangents ; to 
find points of touch, use the counter-circle. 

146. Given the centre, axial directions, a tangent and its point of 
touch P. 

Through P and the intersection of tangent and axis minor draw a circle 
with centre on the axis minor ; it passes through the foci. 

147. Given the centre, a tangent, and 2 a. Use the major circle. 



212 CO-OEDINATE GEOMETBY. 

148. Given the centre 0, a point P of an E, and 2 a. 

Find the corresponding point P'; draw through P a || to axis major 
cutting OP' at B ; OB is axis minor in length. 

149. Given the centre, axial directions, and two points of the conic, 
P P 

Express a^ and IP' through the Cds. of P^ and P<^. 

150. Given two tangents and their points of touch, and the direction of 
the axis major. 

Construct a diameter and apply 149. 

151. Given a point P of an E and the axis minor in length and position. 
Draw II to axis major, through P, cutting minor circle at P" ; the cen- 
tral ray through P" cuts the ordinate of P on the major circle. 

152. Given a tangent and the axis major in length and position. 

153. Given two conjugate diameters, AA', BB' , in length and position. 
Draw a tangent at B ; lay off a' on the normal from B ; through the 

end of a' and the centre draw a circle with centre on the tangent ; it cuts 
the tangent on the axes. 

154. Given any pole P and its polar L, a directrix, and the position of 
the axis major. 

Let L meet the directrix at D; on PD as diameter, draw a circle * it 
cuts the axis major at a focus. 



155. Given the asymptotes and the foci, or the transverse axis. 

156. Given the transverse axis and a point of the H, Ifi — — ^ — 

x"^ — d^ 

157. Given the asymptotes and a point P of the H. 

Draw through P tracts ending in the asymptotes ; from P lay off on 
the longer segment the difference of the two segments ; so are got any 
number of points of the H. Draw the tangent at P II to the fourth har- 
monic to the asymptotes and the diameter through P ; the asymptotic 
Cds. of the vertex are each the geometric mean of the asymptotic Cds. 
of P. 

158. Given the asymptotes and a tangent. 
Halve the tangent intercept. 

159. Given the asymptotes and the difference of a and h. 

160. Given an asymptote, a tangent, its point of touch, and a secon:! 
point. 



EXAMPLES. 213 

161. Given the centre, an asymptote, a tangent, and the ratio a : 6. 

162. Given the centre, an asymptote, and two points. 

163. Given an asymptote and three points. 

164. Given tlie vertical tangents, point of touch of one, and a third 
tangent. 

165. Given two tangents and the focus of a /* ; find the point whose 
focal ray is the half-sum of the focal rays to the points of touch. 

166. Given focus and (1) two points, or (2) one point and a tangent 
of P. 

In (1) draw about each point a circle through the focus ; either outer 
common tangent is directrix ; in (2) either tangent to the one focal circle 
from the counter-point of the focus as to the given tangent is directrix. 

167. Given the directrix and two points (or one point and axis) of a P. 

168. Given the directrix, a tangent, and a point of a P. 

169. Given the vertex, the axis, and a point of a P. 

170. Given the axis, a tangent, and its point of touch (or the vertex). 

171. Given the vertical tangent, another tangent, and its point of 
touch. 

172. Given two tangents and their points of touch, in a P. 

173. Given the vertical tangent and two others, in a P. 

174. Given three tangents (or two points) and the axis of a P. 

175. Given four tangents to a P. Use focal circles. 

176. Sides, altitude, base of an isosceles A are tangents, axis, chord of 
a P. 

177. A P touches one side of a A at its mid-point, and the others pro- 
longed. 

178. Given the axis, the parameter, and a point of a P. 
Draw normal first. 

179. Given the directrix (or focus, or axis), a pole, and its polar as to 
a P. 

Loci. 

180. Find the locus of a point of a tract whose ends move on fixed RLs. 

181. Fixed are the base of a A and the point of touch of escribed 
circle ; find locus of vertex. 



214 CO-OEDINATE GEOMETRY. 

182. The product of ±s from P on two RLs, is k"^ ; where is P? 

183. Given a side and counter Y of a A ; find locus of its mass-centre. 

184. Find the locus of the fifth noteworthy point in a A, given a pair 
of counterparts, or a pair of adjacent parts. 

185. Find the locus of mid-points of chords of an £, through a point. 

186. From two points on a tangent to a circle, d apart, are drawn two 
other tangents ; where do they meet 1 

187. An £ turns about its centre ; where it cuts a fixed E,L. tangents 
are drawn to it ; where do they meet ? 

188. A circle intercepts given lengths on two given RLs. ; where is its 
centre 1 

189. Find the locus of mass-centre of a A of constant area, two of 
whose sides are fixed. 

190. A A is formed by a fixed RL. and two sides of a given Y turning 
about a fixed point ; fiifd the locus of centre of circumscribed circle. 

191. Tangents from P to P form with the polar of P a A of constant 
area ; find the locus of P. 

192. A vertex of a A is fiLxed, the constant counter-side is pushed along 
a RL. ; find the locus of the centre of the circumscribed circle. 

193. The base of a A is given ; the vertex glides on ?/ -f- nx^ = mx, 
whose directrix is II to the base ; find the locus of centre of mass of the A. 

194. Find the locus of pole of tangent to ?/^ = 4 qx as to x^-\- y"^ = r^. 



Homogeneous Co-ordinates. 

195. The Eq. of x cosa+ ij sina—p = in homogeneous Cds. 
(N„ N„ iVg) is v,N^ + v,N, + u,N, = ; find u„ v^, v^. 

We have N-^=^x cos a-^-\-y sin a^—p^ — 0, and so for N,^, iVg; hence, 
Vi cos a^ -f v^ cos Cj -f v^ cos cg = cos o, 
Vy sin ay -f- v^ sin a^ -f Vo. sin a^ = sin a, 

Vi = I COS a sin a^p^]: A, 
j/j = I cos a^ sin ap^ \ : A, 
1/3 =: I COS a-y sin a^p ] : A, 
A = I COS ttj sin KjPg I . 



EXAMPLES. 215 

196. Base lines are Z^^Sa? — 2y + l = 0, L^ — 2x—y — 'd, = 0, 
L^ = X -\- y -{■l — O; find the Eq. of a:+2y — 4 = in homogeneous 
Cds. (Zi, L^, Zg). 

We have 5 A^ + 2A2 + A3= 1, — 2 A^ — A2 + Ag^ 2, A^ — 3 A2 + ^3 = — 4; 
whence, on finding A^, X^, A3, substituting in AjZj4- AgZg + A3Z3 = 0, and 
multiplying by 23, - 20 Z^ + 39 Z^ + 4-5 Zg = 0. 

197. Show that the two Eqs. 

Tj JSf-^ + T2 1^2 + ''■3 -^''3 — ^ ^^^ -^1 ^^^ ^1 + -^^2 Si^ ^2 + -^3 S™ ^3 = ^y 

represent each the EL. at 00. 

198. Where does v-^ N-^_ + ^'2 ^2 + ^z -^z — ^ ^^^ ^^i® sides of the 
referee A ? 

Find the intersection with N-^ — from v^ ^2 + ^3 ^z = ^y "^2 -^2+ '^z -^3 
= 2 A, whence iVg = 2 Ay^ : (t2 ^3 — Tg u,J, N.^ = 2 Av.^: (rg 1^2 - r^ v^), and 
so for the others. 

199. Where do v^ N^ + v^ N^ + v^ N^ = and u^'N^ + U2'^\^ + ^'s'-^^s = 
meet 1 

Since t-^^N-^+ r^N^'i- r^N^=2 A, iV\ = 2 AJTgi'g'l:] j/^ f2''''3lj and so on. 

200. Eind the Eq. of the EL. through (xA^/, iV^2'; ^W) and 
{N,", N^', N,"). 

Assume riiV^ + ^2-^2 + ^3-^3 = ^ j *l^en j//iV/4- j/2iY/+ i/g A\j'= 0, 
V, N,"+ V, N,"^ u, N,"= ; .-.I X, N,'N," \=0. 

201. Eind the ELs. through (xV/, AY, ^^3') and the vertices of the 
referee A. 

202. The ELs. A''i = 0, N^ = 0, u^N^^ u.^N^= 0, p^' ^\ + u^' N^ = Q 
form a four-side ; find the diagonals and where they meet. 

203. Show that the diagonals of a four-side are cut harmonically. 

204. Homogeneous Eqs. of II ELs. differ only by constants. 

205. Eind Eqs. of ELs. through the vertices of the referee II to the 
counter-sides. 

206. Find Eqs. of ELs. through the vertices and the mass-centre of the 
referee. 

207. When are p^ N^ + ^2^2+ ^3 -% = and y^'N^ + u^'A\ + v.^N^ = 
perpendicular 1 

208. Eind Eqs. of the altitudes of the referee and the Cds. of the 
orthocentre. 

209. Find Eqs. of junction-lines of the feet of the altitudes of the 
referee. 



216 CO-ORDINATE GEOMETRY. 

210. Find Eqs. of the mid-perpendiculars to the sides of the referee. 

211. Eind Eqs. of RLs. through the vertices of the referee and the 
points of touch of the inscribed and escribed circles. 

212. Eind the distance of (iV/, iV/, N^') from p^N^-\- u^N^+ j/giVg = 0. 
AVhen but two Cds. are used to fix a EL., call them I and m, and write 

the Eq. of the enwrapt point thus : 

P = ul -\-vm + 1 = 0. 

213. Eind the Cds. of the junction-line of P^ = and Pg^O. 
Proceeding exactly as if to find the junction-point of L^= 0, L^ — O, 

we find 1= {V1 — V2) '• \uiV.^\, vi= — (uj^ — u^) : \u1h\2\. 

214. Eind the distance of P^ = from the RL. (/^ m{}. 

To say the Cds. of the RL. are l^, m^, is to say its Eq. in rectilinear Cds. 
u, V, is l^u + m-^v +1 = 0; to say the Eq. of the point is ii-J, + v-^m + 1 =: 0, 
is to say its rectilinear Cds. are u^, i\, since they fulfil the Eq. of any RL. 
through it ; hence, the distance sought is 

?(,/t + v^m^ + 1 

V/f + m^^ 

215. Eind the Eq. of the point that cuts the tract between Pj^=0, 
and P2 = in the ratio n^ : n^. 

Think of {u^, v^) {u.^, v^) as the points in rectilinear Cds., then 



i\ + «2 "1 + ^2 

is the cutting point ; the Eq. of this point, viewed as enwrapped by the 
varying RL. (/, mi), is 

■ -. ■ / -\ -T TO + 1 = 0. 

Eor Wj and n.^ like-signed, the point is an inner one, otherwise an outer 
one. The Eq. may also be written 

^1^2 + ^2-Pl ^Q 
Wj^ + ^2 

The inner resp. outer mid-point is P^ + P2 = resp. P^ — P^^ 0. 

216. The vertices of a A are P^ =0, P2 = 0, P3 = ; find the 
mass-centre. Erom (215) it is seen to be (P^ + Pg + P3) : 3 = 0. 

217. Three vertices of a parallelogram are P^ = 0, Pg = 0, Pg = ; 
find the fourth. 

218. Eind Eq. of any point on junction-line of P^ = and P2 = 0. 
It is Pj—kP.^ — O; for this is Eq. of a point, being of first degree in 



EXAMPLES. 217 

I, m, and it lies on a RL. through P^ = and Pg — ^j since it is ful- 
filled where they are. From (215) it is seen that h is the ratio of the 
distances of the point from P^ = and P^ = ; varying k gives all 
points on the RL. 

219. When do Pi = 0, P^^ 0, P3 = lie on a RL.? 
Clearly when [ u^r^l 1 = 0, or when /j^Pj + k^P^ + k^P^ — 0. 

220. Show that the points at cx) on the sides of a A lie on a RL. 

221. The product of the ratios in which the sides of a A are cut is 1 ; 
then, and only then, the cutting points are on a RL. 

The cutting points are P^ - k^P^ r^ 0, P, — k^Fo^ =0, P3 — k.J^^ = ; 
the determinant of these three Eqs. vanishes only when k-^ .k^ .k^ = 1. 

222. How do Pi + kP.^ = and P^ + kP^ = lie on the RL. 

223. The cross-ratio of P^ - k^P^ = 0, k=:l, 2, 3, 4, is 



rC-i fCn • nJo fCi . fCn fCo • A-' J rC-i • 

All the problems of the text as to rays may be repeated as to points. 
All the problems in homogeneous point-Cds. may now be paralleled by 
problems in homogeneous line-Cds. E.g. : 

224. Be Pi =0, P^ — 0, -P3 = three vel-tices of a A ; express 
P =: through them in the form KiPi + ic^^P^ -f /C3P3 = 0. See (195). 



225. Writing p— P: V/^ + mi^, show that in Kj p^ -{• K.j.p^-\- K.^p^ = 0, 
the /c's are proportional to the distances of this point from the sides of the 
referee. 

226. Eind Cds. of the RL. through K-^p^ + ^2^2 + HPz = ^ ^^^ 
k'iP^-\- K'2P2 + fc'sPs = 0. 

227. Eind Eq. of junction-point of RLs. { p' i, p' 2> P' s) ^^^l [p"x,p"2,p"^)- 

228. When is K-^p^ + '<"2 7^2 + "^3 P3 = ^ ^ point at 00 1 

The RLs. (Pi,P2,P-i) and {p-^ + d, p^ -f (/, p^ + d) are II ; since they go 
through the same point at 00 , their Cds. satisfy the same Eq. ; or, 

K1P1 + K2P2+ K^Ps^O and K^{p^+d) + K,^{p.^ + d)+ K^{p^ + d) = 0; 

whence /Ci + /(■2 + «"3 = 0. 

229. Show that the mass-centre of the A p^ = 0, ^2 = 0, JO3 = is 
Pi + P2 + Ps = 0. 

230. The centre of the circle about the referee is 

/>! sin 2 ^li -f- jc»2 sin 2 ^12 + p^ sin 2 ^3 = 0. 



218 CO-ORDINATE GEOMETRY. 

231. The orthocentre of the referee is 

p-^ tan A^ + JO2 tan A^ + p^ tan A^ = 0. 

232. Mass-centre, orthocentre, and centre of vertices of a A lie on a 
RL. In the determinant of the Eqs. of the points multiply first row by 
2 sin A^ sin A^ sin J3 ; take from third row ; take out 2 cos A^ cos A^ cos A.^. 

233. The centre of the circle in the referee is 

jOj sin A^ + P2 sin A2 + p^ sin A^ = 0. 

234. The intersection of the transversals from the vertices to the points 
of touch of the escribed circles is 

Pidot^ + p^cot -^ + p^cot^ =0. 

235. The points of (229), (233), (234) lie on a RL. 

236. The point «iPi + «'2P2 + ''3P3 = is distant from the RL. 
ip'vP'vP'z) ('^iP'i + '^2/2 + '^3^3) : ('^1 + «2 + «3)- 

Envelopes. 

237. "When does the RL. {l,m) touch the P whose axis falls on the 
+ X-axis and its focus on the point ql-\- 1 = 0? 

The RL. lu -\- mv -\- 1 = touches if — A^qu when the roots of 

v^ . 1- 7wi; + 1 = are equal, or when rr? ^q — l, which is therefore the 

4^ 

Eq. of the P in line-Cds. 

238. Through a fixed point P is drawn a RL., to which, where it meets 
a fixed RL., is drawn a ±; find the envelope of the ±. 

Take the fixed RL. as F-axis, the ± to it through P as X-axis ; be 
u and V the intercepts of the enveloping RL. ; then v^—pu, i.e., the 
envelope is a P, P is the focus. 

239. Eind the tangential Eq. of E referred to its axes. The intercepts 
are u = a^: x^, v = h^ : y-^; whence, on squaring, inverting, and putting 

J- 4- -iJ- = 1, results — -\ — = 1 ; or, calling the reciprocals of the inter- 

cepts I and m, aH^ + b'^m'^ — 1. Otherwise, thus : 
1 



a 







-i- 

62 



whence K=i , J=—. — ^ C = 

6^ d^ d^h'^ 



0-1 
whence, on substituting and clearing, the same result is got. 



EXAMPLES. 219 

240. About the point {e, o) is drawn a circle with radius 2 a, from 
(— e, o) is drawn a ray to the circle ; find the envelope of its mid-perpendic- 
iilar (a > e). 

The Eqs. of the circle and ray are x — e + y^= 4:a^, y = s{z -{- e) ; 
that of the mid-perpendicular is 

The intercepts of this are 



Va^(l + s'^)— eV^ = M and Va-{1 + s-}—e^s- : s= v; 
whence eliminating s and putting /, m for 1 : u, 1: v, we get 

a2/2+(a2_e2)r/i2=l: 
the envelope is an £. 

241. Through {e,o) and(— e, o) in the circle x^ + i/^ = a^ are drawn 
II chords; find the envelope of the RL. joining two ends of the chords on 

the same side of a diameter. a}p -{-{a^ —e'^)7n'^= 1. 

242. The II s y = ci, y = — <^, meet the circle x- + y- —2xx— e^, 
and the points of meeting are joined crosswise ; find envelope of junction- 
line when A varies. ci-in- + (a^— e^)!^ =. 1. 

How does the envelope change as e changes % 

2 2 

243. The pole, as to ^+^=1, traces the circle ar^ + y^ — (j2; 
what does the polar envelop ? Ans. aH--\-hh-n?- = a-. 

HixT. Eq. of the polar is — ^+ ^l^= 1 ; here -i= /, ^i= m. 

a- b- cr h' 

244. Eind the envelopes when the pole traces x"^ -\- y- = h- and ci^ + h"^. 

245. Eind the envelope of the junction-line of the ends of two conju- 
gate diameters. Ans. The E a-P + b-m'^ = 2. 

HixT. The ends are the points (a cos e, b sin e) and (— a sin e, b cos e) ; 
the EL. through them is 2-6(sin e — cos e)— _ya(sin e-f cos6)+ a6 = 0; 
here (cos e — sin 6)= a/, (cos e -f sin6)= i;?i ; hence the above result, 
on squaring. 

246. Erom the point {e, o) ravs are drawn to the circle 

{x+ey- + y'=--a^; 
find the envelope of their mid-perpendiculars. 

247. The vertex of a right X glides on x"^ + y^ = r^, one side en- 
wraps the point (e, o) ; what does the other side enwrap ? 

Ans. r-l'^ — (e^ — r-)m^ = 1. 



220 CO-OEDINATE GEOMETRY. 

248. What is the tangential Eq. of 7 x"^ — 5 f + 12 x -{- 8 1/ — ^7 = 1 

249. Show, in two ways, that the tangential Eq. of f/ referred to its 
asymptotes is lm{a'^-\-U^) = 1. 

250. When does the general tangential Eq. of second degree, 

KP + 2 Him + Jm"^ + 2 Gl + 2 Fm + C =0, 

represent an £ ? when an // ? when a /* ? 

Be kx^ + 2 hxt/ +jt/'^ + 2gx + 2fij + c = the Cartesian Eq. from 
which the tangential is got by putting for k, h, etc., their co-factors K, H, 
etc., in A ; also suppose A > 0. Then, as both Eqs. picture the same 
curves, the criterion is the same for both : the curve is £, P, H, according 
asC>0, C = 0, C<0. 

251. Putting k', h', etc., for the co-factors of K, H, etc., in A, show that 
the tangential Eq. pictures two points when A = and c' < 0, pictures 
one double point when c' — 0, h' — 0, k' — 0. 

252. Discuss 250 and 251 geometrically, remembering that from every 
point of the RL. at eo may be drawn two tangents to £ ; only from outer 
points may they be drawn to H; from every point may be drawn only one 
tangent to P, since the RL. at 00 itself touches P; and combine with the 
given Eq. of second degree the Eq. I — \m = of a point at 00 . 

253. From a point on the X-axis are dropped _Ls on the RLs. x = y 
and a: + 2 3/ = 10 ; find the envelope of the junction-line of the feet of 
the -Ls. Ans. A P. 

254. Two RLs. mutually ± turn about a fixed point; find the envelope 
of the junction-line of their intersections with two fixed RLs. 

255. Through {o,d) is drawn the secant x-\-y=^d of the system of 
circles x"^ -\- y"^ — 2 Xx = cP' ; find the envelope of the tangents at the 
points of secancy. 

256. Through (0, d) are drawn secants to x"^ -\- y"^ — r^, and ±s drawn 
to the secants at the points of secancy; find their envelope. 

257. Find the envelope of the polars of a point as to a system of con- 
focal conies. 

258. A secant cuts a system of conf ocals ; find the envelope of tan- 
gents at the points of secancy. 

259. The two points at ao in : H are real; P, coincident; £, imaginary. 

260. Two As, one formed by tangents to a curve, the other by chords 
joining the points of tangence, may be called outer resp. inner, and said to 
correspond. Show that in P the outer is half the inner. 



EXAMPLES. 221 

261. Establish Carnot's TJieorem : The product of all the ratios in 
which a conic cuts the sides of a closed polygon is 1. 

Hint. By Art. 72 the product of the ratios in which the conic J^= 
cuts the side JP1P2 is ^1 : F^. 

262. If i^i = k^x^ + 2 h^xy + j^ + 2g^x-[- 2f^y + c^, and A be a 
parameter, then F^ + xF^ = is a system of conies : through how many 
points ? what pairs of KLs. belong to the system 1 what P's ? when does a 
circle belong to it ? what is the locus of the centres "? how lie the polars of 
a point as to the members of the system 1 how lie the poles of a given 
EL.? 

263. Show that any EL. cuts the system in a system of points in Invo- 
lution, whose /bc2 are the points cutting harmonically the chords of the base 
conies. 

264. Show that when the pole traces a EL. the perpolar envelops a P. 

265. Eind the envelope of normals to an £, an H, a P. 

266. Eind the envelope of the perpolar when the pole traces a conic. 



Part 11. Of Space. 



This subject is much more extensive than the Geometry of the 
Plane ^ so that any detailed treatment here is out of the question ; 
onh' the most essential notions can be developed. At the same 
time, the close analogy of the two doctrines permits a much 
more condensed discussion than was possible in Part I. 



CHAPTER I. 



1. We say of Space, it is triply extended or has three dimen- 
sions, meaning that three determinations are needful and enough 
to fix any element of it. These determinations may be made 
in many ways, giving rise to as many systems of determining 



Y 
C 




-X ; // / 'A +x 



y I 



-y 

-z 
magnitudes, Co-ordinates. Thus, suppose three planes meeting 
at O; call their intersections the JT-, Y-, Z- axes, the planes 



224 CO-OKDINATE GEOMETRY. 

themselves the YZ-, ZX-, XY- planes : when no confusion would 
arise, omit the words axes and planes. A plane |1 to YZ, cut- 
ting off a tract OA = a on X, has all its points at a dis- 
tance a from YZ measured I| to X, and no other points are so 
distaiit ; hence it is defined completely by the Eq. a?=a. So, 
too, 2/ = ft, z = c, are Eqs. of planes \\ to ZX^ XF resp. 
The first pair meet in a RL. || to Z, for all points of which, and 
for no others^ the relations hold : sc — a^y—h^ which are there- 
fore the Eqs. of a RL. \\ to Z. 

So, too, y = l^i z = c resp. z = c, x = a are Eqs. 
of RLs. 11 to X resp. Y. As a special case, 07=0, y = 0, 
z = are the Eqs. of YZ, ZX, XY\ 2/ = 0, ^ = 0, and 
2; = 0, 07=0, and £c = 0, 2/ = are the Eqs. of X and F 
and Z. The three planes and three RLs. of intersection meet in 
a point for which, and which alone, hold all three relations : 
x — a, y = l>9 z — Cf which are therefore the Eqs. of the point. 

It is most convenient to think XY horizontal, right or east 
being -{- X, forward or north + Y, as in Plane Geometry. 
Either up or down may be taken as -f- on Z, but up is better, 
according to the convention, important in Mechanics : That side 
of a plane is + whence positive rotation (as from -j-^to -f- Y) 
appears counter-clockwise. 

Clearly Space is cut by the three planes into eight regions. 
The upper four we name 1, 2, 3, 4, from the quadrants in XY 
, , on which ihey stand ; those below, in the 

"T" ^ "T" 5 -1- 

same order, 5, 6, 7, 8. 



+ , ± 



+ ' ~~' i Then the signs of x, y, z in the eight 
regions are, as in the diagram, the loioer sign of the z referring 
to the lower region. 

The ^s yz, zx, xy may be denoted by x-,^-,^'-, unless otherwise 
stated they will be considered right ^s. We maj^ call x, y, z 
triplanar Cds., and speak of the ^om^ (x, y, z). 

2. Around any RL. (say Z) as axis, suppose laid a cylinder 
of radius r = ri. AH points of the surface, and no others, 
are distant ri from the axis, and the surface is defined completely 



SPHERIC CO-ORDINATES. 



225 



b}^ its Eq. r=r^, Thi^oiigh the axis pass a half -plane sloped 
= $1 to some &ase-plane through the axis (say ZX) ; then 
is this half-plane defined completeh^ by its Eq. , 6 = 6i. For 
all points on the E-L., the intersection of half -plane and cyl- 
inder, and for no others, hold the relations : r = 7\, =.0i, 
which are therefore the Eqs. of that ML. Pass a plane _L to Z, 
hence 1| to XY,. and distant z = Zi from this latter. By 
Art. 1 , z = Zi is its Eq. For the intersection of this plane 
and the RL.' (?^i, Oi) , and for no other, hold the relations r — i\, 
= 6i, z = Zi, which are therefore the Eqs. of the point. 
We may call r, 0, z cylindric Cds. , and speak of the point {r, 0, z) ; 
rmay always be taken -f-? 0-\- when reckoned counter-clockwise, 
z -\- when reckoned up. Connecting the two systems of Cds., 
the relations hold : 

00 — r cos 9, y = r sin 6, z = z. 



3. About any point (say the origin 0) lay a sphere of radius 
pi ; clearly its Eq. is p— p^. Pass a half-plane as in Art. 2 ; 
the Eqs. of the half great circle in which the half -plane meets 
the sphere are clearly p = pi, = 0i. 

About Z lay a cone sloped S^ to Z and (jf)i to XY, so that 
8i + (^i = 90°; its Eq. is S^Si, or (/> = c/)i. For the 
point where it meets the half-circle, and for no other., hold the 

Z 






in 



yX^-'-^J 



^ 



X 



Y 

relations : p = pii 
therefore its Eqs. 



(j) =z (f)^ (or 3 = Si) , which are 



226 



CO-OEDINATE GEOMETRY. 



We may call p, 6, <^ (or S) polar or spheric Cds.^ and speak 
of the point (p, 0, cfi) ', p may be taken always +, -\- when 
reckoned counter-clockwise, <{> + where reckoned toward -j- ^ 
(8 always +) . Clearly 6 and ^ correspond to geographic Longi- 
tude and Latitude : S may be called (north) polar distance. 

On projecthig p on Z and on XY, and this last projection on 
X and T', these relations become manifest : 

x = p cos (^ cos ^ = p sin 8 cos ^, 

2/ = p cos (fi sin = p sin 8 sin 8, 

2: = p sin (jf) = p cos S. 

4. Hereafter, cosine resp. sme may be denoted by putting a 
horizontal bar under resp. vertical bar after the argument, thus : 

w\ = sin CO, 00 = cos to. 

Call the tract OP from the origin to any point P the radius 
vector of the point, and denote it by p ; denote its slopes to 




X 



X, Y", Z by a, /?, y, and call their cosines a, /8, 7 direction-cosines 

of p. Then, by definition, the projections of p on the axes are 
the Cds. of P; i.e., 

X=ap, y = (3p, Z = yp. 

Squaring, adding, and remembering x^ -\- y'^ -\- z^ — p^, we get 

a2 + ^2-f/=l. (A) 



DIEECTIOISr-COSINES. 227 

For one factor in each term put '-, ^, -, multiply by p, and get 

P P P 

ax + (^y + yz = p, (B) 

which simply says the sum of the projections of a train of tracts 
from O to P on OP is OP, as is already known. 

Now take a plane perpendicular to OP ; it will be sloped a, ^, y 
to YZ, ZX, XF; let it meet X, F, Z at A, B, (7, and call the 
area of the A ABC A ; the projections of this A on FZ, ZX, 
XT Sire the A BOC^ COA, AOB ; their areas are aA, ^A, yA ; 

squaring, we get aA +/8A +7A'' = A^ since a^4-iS^ + y^=l; 

i.e., the squared hypotenuse-face of a right-sided tetraeder equcds 
the sum of the squares of the other faces. This proposition is the 
analogue (for space) of the Pythagorean. 

The distance d between P^ (p^, aj, /5i, y^) and P^ {p^-, ^21 (^2-> 72) 
is plainly the diagonal of a parallelepiped, whose edges are 

.1 /72 _ ~; 2 ■ 2 2 

U/J ~~ '^21 y\ — 2'25 1 — 2 ? nence a — xj — C02 ~7~ 2/1 — ^ 2/2 i* 1 — -2 • 

By the iaio 0/ Cosines d^ = pi 4- P2^ — '^pip2pip' 
Hence, 

P1P2 = (a^i^2 + yi2/2 + ^1^2) : P1P2 = aiag -h ^1^+7172 (C) 

This last expression for the cosine of the ^ between two RLs. 
in terms of their direction-cosines holds even when the RLs. do 
not meet, since the '2f between two non-intersecting RLs. equals 
the ^ between two || intersecting RLs. 

CoROLLAKY. Whcu^ and only when., pi and pg are J_, 

0-10.2 -{-§1^2 + 7172 = 0. (Ci) 

The ^ between two planes equals the (adjacent) ^ between 
two JLs on the planes, but the sloj)e of a RL. to a plane is the 
complement of the slope of the RL. to a _L on the plane ; hence, 
the slopes to the axes of a plane J_ to p are A = 90^ — a, 
B = 90° - /5, r = 90° - 7 ; hence, 

A|2 + B|2 + rp=l and lOl2 = Ai| A^l +Bi]B2l +ri|r2| (C2) 



228 



CO-ORDINATE GEOMETRY. 



4. In case of oblique axes we use the theorem : the projection on any 
RL. of a tract between two points equals the sum of the projections of any 
train of tracts between the points ; hence 



/--^ 



p . pi = X . xl -\- y . Ill -\- z . zl. 
Take as /, in turn, the vector p and X, Y, Z; so we get 



/--> ^~^ ^-~^ 

p^x.xp+ 7j .yp + z.zp, 



p.yp = X. XII + ?/ + 2 



p .xpr= X ■\- y . yx -^ z . zx 



(D) 



-.'/, 



p .zp zn X . xz -\- y .yz -\- z. J 
On multiplying in turn by p, x, y, z, and adding, there results 
p2 = :3^2 _j_ ^2 _j_ ^2 ^ 2 ya: , yz -\- 2 zx . zx ■{- 2 xy . xy. 

Hence xp, yp, zp, or a, $, y, are found at once by using (D). 

To express conversely x, y, z through p, a, fi, y, form the determinant A 

^~> ^~-\ ^~\ 

of the four Eqs., remembering yz =iXj ^^ = ^, xy — oo ; 

A = — 1 a 3 T , and put S^ — 



1 


a 


3 


2 


a 


1 


CO 


!!f 


^ 


CD 


1 


-^ 


7 


^ 


X 


1 



1 


CO 


i 


CO 


1 


X 


^ 


X. 


1 









then, denoting the co-factors in A by like letters accented, 

x=aV:5^ y^^'p:S\ z = y'p:S\ 

On putting these values in the first Eq. of (D) and clearing, there 
results 

x\^a^ + ^P\^0'+co\^y^-2x"By-2^y'ya-2c^''a0=S^ (A*) 

where x"? ^'> ^" are co-factors of like letters in 5^. 

To find the distance d between two points, P and Pj, take P^ as a new 
origin (see Art. 6), then the Cds. of P are x — x^, y — yy,z —z-^', put these 
for x, y, z, and d for p in the formula found. 

To find the cosine of the Y POP^, put p^ for /; then 

^~X /"^ /-~^ /--X 

p.pp^ = x.xp^ + y.yp^ + z.zp^; 
on substituting for the cosines on the right, there results 

On comparing (A) with (A*) and (C^) with (Cj*), analogy would sug- 
gest the following Eq. as (C2*) : 



PROJECTIONS OF A TRIANGLE. 



229 



^-^ 



- ip"{ya^ + a7i)- a/'(a&^ + iSaJ : 5". 



(C,*) 



This conjecture is readily confirmed thus: on putting, in (Cj^*), for the 
Cds. their values found above, the product pp^ vanishes, 5* becomes 
divisor, subscribed and unsubscribed letters {a, j8, 7) combine every way in 
sets of two, and the result, symmetric as to the subscribed and unsub- 

scribed letters, since pp^^ — p^p, is of the form 

{Aaaj^ + B0^^ + 0771 + A' {^y^ + yB^) + B' {ya^ + ayj) 

where A, B, C, A', B', C depend only on x, ^> <^- 
^~x 
For ppi= 0, this must pass over into (A*), for then a= a-^, etc. ; 

hence A — x\'^-S-, etc., which on substitution yield Cg*, as guessed. 
We have seen that in case of rectang. axes 

Boc'' + coa" + Job'' = abc\ 

i.e., the squared area of a A (and hence of any plane figure) equals the sum 
of the squared areas of its projections on three rectangular planes. To find the 
general theorem for oblique axes, of which the above is a special case, we 
note that the corresponding formulae for rectang. and oblique axes are, 
when -^AOB=o:, 

AB^='OA-+OB- and TB' =ljA~ + '0B'' -2 OA . OB .<^. 

On putting OA = a, OB = b, these formulae may be written 



^J5 =- 






a 


h 


a 


1 





h 





1 



AB'^- 







So, too, putting OC =c, we have in case of rectang. axes 



^ABC ^- 



be ca ah 

6c 1 

ca 1 

a6 1 



whence 4lABC = 





be 
ca 
ab 



be 


ca 


ab 


1 


CO 


!^ 


w 


1 


X 


"t 


^ 


1 



if the analogy holds, a result easily verified. 

The area of a parallelogram whose sides, a and b, are sloped oj is ab sin ca, 



or 



ab 



230 CO-ORDINATE GEOMETRY. 

Accordingly we might suspect the volume of a parallelepiped whose 
edges a, b, c are sloped co, %, \p to each other to be 

abc 



1 


cv 


!^ 


CO 


1 


^ 


^ 


X 


1 



This conclusion from analogy is readily verified thus : take the paral- 
lelogram a6 . co| as base, project (7 on AT at A' and on XY at C (the edges 
being taken as axes) ; calling the diedral Y along X A, we have 



By spheric trigonometry A \ = Vl— x^ — i//'^ — w'^ + 2 x^/^w : ;// ] . co | ; the 

volume is ab.c>}\. CC, whence the formula above. 

The radical of the determinant, which is S of A*, is thus seen to be the 
volume of a parallelepiped of unit edges, sloped w, x, ^ to each other. Since 
the factor 5 turns the product of the edges into the volume of the parallele- 

piped, just as sin ab turns the product of the sides into the area of the 
parallelogram, it has been named (by Staudt) sine of the solid angle of the 

edges, and may be written sin abc or abc \ . 

If g, T], 5 be the slopes of X to YZ, Y to ZX, Z to XY, then 

Ca = c.^\; 

/"^ ^-^ 

but CC = c .y\i\.A\=c.xyz\'. oo\; hence xyz\ — Q.Q}\, or 

5 = ||.xl=^I-^I = tI."I. 

To find the area of any A (or other plane figure) in terms of its projections, 
^~-\ ^-^ ^-^ 

call these projections I .yz\, J.zx\, K .x!j\ {ov I .x\, J •'^\y X.ca\), and 

suppose the A. p on the plane of the A directed by a, 13, y; then 

A.a = 7x|-ll> or A.a = J.S, 
since each is the projection of A on a plane JL to X. Hence 

A.o = /.S, A.^=J.S, A.y^K.S. 

If h be the altitude and I the edge of a prism standing on A as base, and 
i, j, k be the projections of I on X, Y, Z, then A = m + jp + /c7 ; hence 

A.h — volume of prism —{iI+jJ-{- JcK)S. 

The projections of a tract on the axes, II to the Cd. planes, resp. the pro- 
jections of a, plane figure on the Cd. planes, II to the axes, are called Cds. of 
the tract resp. plane. 



VOLUME OF A PYEAMID. 



231 



Now take the pyramid whose vertex is 0, whose base is PP^P^ ; it is |^ 
of the prism with base OP^P^ and edge OP ; the Cds, of the edge and the 
double Cds. of the base of this prism are x, y, z, and 



Vi H 


xl, 


^1 


^1 


'fl, 


^1 Vx 


y% ^2 




^2 


^2 




^2 ^2 



Hence by the above formula we have 



6 OPP^P^ = 



X y 

•^1 Vl "^2 

To move the origin to {x^, y^, z^), it suffices to put x -^ x^ for x, y — y^ 
for y, z —z^ for z, etc., and write 



S. 



^0PP,P,^6PP,P,P, 





1 



j:,,- 



y -Vz 

Vi-Vz 

Vi-Vz 

Vz 



On adding the last row to each of the others there results 



6PP,P,P, 



X 



y 
yi 
y-i 
y-i 



S = 6T. 



When, and only when, this six-fold tetraedral volume is 0, does the 
-point P{x, y,z) lie in the plane of P^P^P^; hence ^^O is the Eq. of 
the plane through the three points P^, P^, P^. 

5. By projecting a point I| to Z (say) , its X and Y are not 
changed; i.e., the x and y of a point are the x and y of its 
Xy-projection, and are the same for all points of a RL. || to 
Z, To find, then, the x and y of a. point cutting a tract P1P2 in 
ratio 72i : ng, project the tract on XY; the Cds. of the projection 
are the Cds. sought : 



05 = 



n,oo,±n^ ^ n,y, + n,ij, ^^^ ^^ 



ni+ n^ 



n^ + ^2 



z = 






Transformation of Co-ordinates. 

6. Fov pushing the axes, not changing their directions, clearly 

x = x'-\-x^, y=y' -j-y^, z = z' -{- Zq. 



232 CO-ORDINATE GEOMETRY. 

If the axes (rectangular) be turned from the position X, Y, Z 

into the position X', F', Z', so that the ^s XX\ XT', XZ', are 
a', a", a'", then the sum of the projections of x\ y\ z' on X, for 
an}^ point, is simply the x of that point ; or 

and so y = 3c'^'-^y'^"+z'^"', >■ (E) 

z = x'y_'+ y'i'-V z'y^"' ) 

The nine ^s are clearly not all at will, since, if a' and f^' be 
chosen, y' is thereby fixed ; for all RLs. sloped a' to X lie on a 
cone about X, and all sloped /5' to F on a cone about Y, and X' 
must be a common RL. of these two cones ; the axis X' being 
fixed, so is the plane Y'Z', and the choice of one more ^ fixes 
Y' and Z'. So 07ily three of the nine can be chosen at will ; 
hence there must hold six Eqs. of condition among the nine ^s. 
These are 

^ni2_^^,f,o_^yii2^l. a"'^a'+§"''/3'-}-f''y'=0. 

The first three say X, Y, Z are rectangular, the second three 
say X', y, Z' are rectangular, as appears from (A) and (B). 
The formulae for passing from X', F', Z' to X, Y, Z are 

plainly 

x'=xa'-{-y/3' + zy', 

yf=Xa"+yl3"+Zy'\ 
z' = Xa"'+y§"'-{-Zy"'. 

Accordingly there must hold these six Eqs. of condition: 

y" -\-y"'+y""=l ; 7'-^'+ f 'a"-\-y"'-a"'=0. 

These six Eqs. must then be equivalent to the first six ; this is 
clear geometrically, and may be proved analytically thus : 



TKANSFOPvMATION OF CO-OKDINATES. 233 

Form the determinant 
C = 

§' ^_" 13'" 

'/ y" y'" 

by solving the first three Eqs. directly, we get 

C'X'=A'x-{-B'y-}-T'z, and so on ; 

hence, A'=a'(7, B'=/3'C, and so on ; now 

a'- A'+ a". A"+ a'"- A"'= (7, and five other like Eqs., 

while a'. B'+ a". B"+ a'"- B'"= 0, and five other like Eqs. ; 

whence, on replacing the co-factors, A', etc., result the twelve 
Eqs. 

By squaring C according to the Multiplication Theorem of 
Determinants, it is shown that C'-=l; hence 0=±1, 
according, namely, as X'Y'Z' is congruent ov symmetric vfiXh 
XYZ ', i.e., according as, when +X' falls on+Xand+I^' 
on + Y, +^' falls on -\-Z or -Z. 

The formulae (E) hold even when X', Y', Z' are not rectan- 
gular, since this rectangularity was not assumed in their 
deduction. 

The general formulae for oblique axes are found, precisely as 
in Plane Geometr}', to be 

a? . (cc, yz) I = oc\ [x', yz) \ + y\ {y', yz) \ + z\ {z', yz) \ , 

and two got by permuting cc, ?/, z. The nine coefficients are 
again connected by six Eqs. of condition. 

Note that the Eqs. of Transformation are linear in Cds. 

EXERCISE. 

Show that p^2 1 ' = I «i^2l ' + I ^i72 i ' + 1 7i«2 1 '• 

The Plane and the Rig-ht Line. 

7. A single Eq. in x, y, z represents a surface. For we may 
assign all real pairs of values to x and y (say) , and reckon 



234 CO-ORDINATE GEOMETRY. 

the corresponding values of z. The pairs (x, y) fix points in 
Xy ; over each such point suppose the fixed vahie of z erected, 
laid off parallel to Z ; the ends of all such z'& will lie on and 
determine the surface, which of course may be real or imag- 
inary, continuous or discontinuous. 

Two Eqs. in x, y, z represent a line. For all points whose 
Cds. satisfy both Eqs. must lie on both surfaces picturing the 
two Eqs., hence must lie on the intersection of the surfaces, 
i.e., on a line. 

Three Eqs. in x, y, z of mth, nth., resp. pth degree^ represent 
mnp fixed points. For the three Eqs. are fulfilled at once by 
mnp triplets of values, each pictured by a point. 

Transformation of Cds. does not change the degree of the Eq. 
in ic, ?/, z. For the Eqs. of transformation are linear. 

8. A line is fixed by the Eqs. of any tioo surfaces through it ; 
the simplest surfaces are generally two cylinders \\ each to an 
axis. The Eqs. of these cylinders are the same as the Eqs. of 
their intersections, each with the plane of the other two axes, 
since the third Cd. is the same for every point of any given 
element of one of them. They are clearly the projecting cylin- 
ders of the line, and their intersections with the planes are the 
projections of the line on those planes. Hence, as Eqs. of a line 
may be taken the Eqs. of any tivo of its projections., on the planes 
of two axes, |1 to the third axis. 

If the line be a RL., the projecting cylinders are planes., and 
the projections are RLs. whose Eqs. may be written 

y=^sx-{-h\ y = tz-\-b., x = uz-\-a, 

any ttvo of which (generally the last) may be taken as Eqs. of 
the RL. Clearly, a, 5', b are the intercepts of projections on 
X, Y., while s, i, u are direction-coeflScients. 

Symmetric Eqs. of the RL. may be got thus: Be (x^^y^, Zj) 
and (x, ?/, z) a fixed and a variable point on the RL., d their 
distance apart, and let its direction-cosines be a, y8, y ; then 



EIGHT LINE IN SPACE. 235 



or x==:Xi-{- ad, y ■=y-^-\- /3d, z = Zi-^ yd. 



In 



x — Xi ^ y — yj _^ z-Zi 
a' ft Y 



instead of a, ft, y, may be put any three proportionates, as X, {x, 
x — xj_ y-yi z-z^ 



V, so that 



X 



fi 



If X:a = fjL: ft = v:y=f, then f= V\^ + p,^ + v^. 

On comparing the symmetric and the tangential forms, we see 

S = ft : a, t = ft:y, u = a : y ; 
whence u= t : s. 

If the RL. goes through (iCg, 2/2? ^2) 5 then 

^2 ^1 Vs — yi ^2 ^1 . 

a ^ y ' 

i.e., ^2 — Xi, 2/2 ~ 2/1? ^2 — ^17 ai'e proportional to a, ^, y ; hence 

^ — ^1 ?y — 2/1 ^ ~ ^1 
= ^ = 5 

X2 Xi 2/2 2/1 "^2 ^1 

Eq. of a RL. through ttuo points. 

9. Two RLs. in space may or may not meet ; if they do, the 
z of the intersection of their projections on YZ must be the 
same as the 2; of the intersection of their projections on ZX : 
i.e. , if y = tiZ-\- bi, X = u^z -f- %, and yz=t2Z-\- 1)2, 

X = U2Z -\-a be the RLs. , then 



\ — &2 • ^1 — h = (^h — ^h ' '^h — '^'■2- 

Or the four Eqs. must hold at once ; this yields the same 
result in a determinant. 



236 CO-OEDINATE GEOjVIETKY. 

If two RLs., ?i and Za? are directed by a^, ^^i, 71 and ag, ^2-> 725 
or by their proportionals Ai, />ti, vi and A2, //-25 ^'2? then 

C" ,00, '^I'^S + /^l/^2 + V1V2 

/i^2 = aia2 + ^ii«2 + yi72= , — , f == 

1 + ^1^9 + '?^1^^2 

I ^^ — ^ — » 

If the RLs. be perpendicular, then 

A1A2 + /^l/^2 + I^ll'2 = = 1 + ^1^2 + '?^l'^2« 

If the RL. (^, w) be _L to {t\ u') , then 

l+tt'+iiu'=0; (1) 

if it goes through (xi, 2/1, 2;i) , then 

2/1 = ^^1 + ^5 a7i = ?^;2i + a; (2) 

if it meets the RL. (t', w'), then 

b — b' : t — t'= a^^' : ic — u'. (3) 

These four Eqs. fix the values of t, u, a, b. They may be 
found thus : 

From (2) , a = Xi — uZi, 6 = 2/1 — ^% ; put these into (3) , 
whence 
(u'zj + a'— x^)t — {t'z^ + V— yi)u = (a'— o^i)^'— (b'—7j{)u'. (4) 

From (1) and (4) can now be found t and u, thus : 
Put 2^= u'{u'z^ + a'— cci) + t'{t'z:^ + 6'- 2/1) , 

M=t'l(b' — yi)u'—(a'—x{)t'l — (u'z^-\-a'—Xj), 
L =u^\{a^-x^y-{V-y,)u^\-{t'z^+b'-y^)', 
then t =M:]Sr, u = L:N] 



L . M 

a —x^ — — z^, o = yi — —-Zi. 



Accordingly, 



X — Xi'. L = y — yi'. M= z —z^: N 

is the Eq. of the ± from (x^^ 2/15 %) on the RL. y =tz-\-b, 
x = uz-\-a. 



PEEPENDICULAE TO TWO BIGHT LINES. 237 

The student may now easily show that the Cds. of the inter- 
section are 



2/i-(lf:l+i'^ + ^*'^), 



while the distance from (xj, 2/1, z-^ to the intersection is 
or, after simplification, 



10. If the RL. y = tz-\-h, x =i uz -}- a, be X to the two 
RLs. 

yz=tiZ-\- bi, X = UiZ + %, 

and y = t2Z-{- 627 ^ = '^2^ + ct2) 

then ^1^ 4- '^i^ + 1 = 05 ^2^ + '^2'^ + 1=0, 



whence ^ = w^ — ^2 : ^i^i2 — ^2'^ii u=:ti — ^2 • '^1^2 — '^2^1' 

If the RLs. be directed by (a, /3, y), (ai, ^1, yi), (ag, /327 y2)5 
ctitt + ^1^ + yiy = 0, 

ttsa + ^2;^ + 727 = 0? 
a' + ^'4-/=l. 

From the first two Eqs., on solving as to a : y and ^ : y, it 
results that 

a : ^ : y = ^^ys — ^27i - 71^2 — «i72 • "lA — ^i(h- 

If the RL. (if, u) also meets the RLs. (^1, Ui), (^25 '^2)? then 
(b — &i) (li — Ui) = (a — ai) {t — t^) 

and (& — 62) (^ — '^2) = (ct — a2) {t — ^2) . 

/ 



238 CO-OEDINATE GEOMETRY. 

Form the determinant 



1 


1 


1 


u 


III 


U2 


t 


k 


h 



and denote its co-factors, as usual, by like large letters ; then 

and so for aD. 

Thus is determined the common _L to two RLs. 
The length of the intercept on it between the two may now 
be found by finding the intersections ; but that is tedious. 

11. The S3"mmetric Eqs. of a RL. . -^= ^ — ^ = ~ ^ , 

X fx y 

contain six parameters or arbitraries^ A, /x, v, Xi, y^, z^, which 
might be called the Cds. of the RL. in Space. But they are not 
independent^ since four arbitraries (Cds.), (t, u, a, 5), fix the 
RL. First, A, /x, v are proportional to a, /?, y, and these are 
connected by the relation a^ + /^^ + y^ = 1 ; secondly, denot- 
ing jxZi — v?/i, vXi — X^i, Xyi — fxXi by A, M, N, we see the 
relation holds : 

XA'+/xM + vN = 0. 

These six symbols, X, /x, v, A, M, N, thus connected by two 
Eqs. of condition, we may call Cds. of the RL. 

The last three are interpreted geometrical^ later. (Art. 25.) 

12. The Eq. of a plane is of first degree in x, ?/, z. For the 
Eq. of a plane || to a Cd. plane, as XY, is 2 = %; this plane 
may be referred to any other system of axes by a linear trans- 
formation of Cds., and such a transformation cannot change 
the degree. 

Accordingly the Eq. of a plane is of the form 

lx-\-my + nz -^d = 0. (1) 

If a, &, c be the intercepts on the axes, then clearly they 



THE PLANE IN SPACE. 239 

equal —d:l, —dim, —d:n; hence they vary invei^sely 
as I, m, n; the Eq. of the plane msLj also be written 

- + 1 + ^=1. (2) 

Conversely, an Eq. of first degree in x, y, 2, represents a 
plane. For, if d be not 0, it may be brought into form (2), 
which is known to represent a plane making intercepts a, 6, c 
on X, Y., Z., if d be 0, push the origin out any distance, say d', 
on any axis, say X, by putting x + d' for x in the Eq. ; then 
the Eq. may be brought into the form (2), and the previous 
reasoning applies. 

13. Drop a _L ^ from on the plane, directed by a, ^, y ; 
then p = aa = bl3 = cy, and on substitution results 

xa + yf3-hzy—p = 0, (3) 

the Normal Eq. of the plane, which we may write 

If F be the factor that turns the general into the normal 
form, then 

Fl = a, Fm = f3, Fn = y'/ 

.\F\P + m'-i-n') = l, 

whence F=l : V?^ + m^ -f 7i^, 



and a = l : Vl^ + 7n^ + n^ ; 

and so for ^ and y. 

13.* In case of oblique axes we have 



abc .S = p ^/{ab .(t}\ -{■ be . x\ -{-ca.xpl 

~2(ab.a:\.bc.x\Y+bc.x\-ca.xl^].Z+ca.yp\.ab.a}\X)}, (4) 

since each is the six-fold volume of the tetraeder 0—ABC, where 
OA = a, OB =b, 0C= c, and X, Y,Z are cosines of the diedral Ys 
along X, Y, Z. Call the As BOC, CO A, AOB the p/anar intercepts of the 



240 



CO-ORDINATE GEOMETRY. 



plane whose axial intercepts are a, h, c, and denote them doubled by 
A,B,C', then 

aic.5 = jo VM'+^'+ C^-^AB.Z+BC.X-{■ CA.Y)}. (5) 



In (4) put for a, h, c, p their values , 



i 

VI 



Fd ; hence 



-2{I.x\m.^\.Z+m.rp\.n.cv\X+7i.c^\l.x\Y)}. (6) 

The analogy of this to the value of F in plane geometry becomes plain 
on writing each in determinant form : 



F^ = 





I 
m 



m 


. F2^ 


1 


CO 


"k 




CO 




W 


1 


^ 




1 




i 


^ 


1 





m n 

0) \p 



m CO 



n ^ X 







By reasoning like that in plane geometry it is now shown that the dis- 
tance of {x, y, z) from the plane xa + y^ + zy—p — Q is xa-\- y^-\- zy—p, 

or {x, y, z) is distant N{x, y, z) from N{x, y, z) = 0. 

The whole body of reasoning as to normal Eqs. of RLs. may now be 
repeated as to normal Eqs. of planes ; and as there the Abridged Notation 
issued in a system of homogeneous triangular (or trilinear) Cds., so here it 
issues in a system of homogeneous tetraedral (or quadriplanar) Cds. ; and 
just as the first could also be interpreted as line-Cds., so the second can 
also be interpreted as plane-Cds., a thought that cannot be developed here. 

14. We have found the Eq. of a plane through three points 
and the six-fold volume of a tetraeder, given by its vertices, to 
be respectively 

= 0, and 6 T= 



X 


y 


z 


1 


x^ 


Vi 


^1 


1 


X2 


2/2 


^2 


1 


x^ 


Vs 


% 


1 



X 


y 


z 


1 


x^ 


2/1 


2^1 


1 


X<i 


.% 


2^2 


1 


Xs 


2/3 


% 


1 



s. 



(8) 



These two Eqs. are really one, the first merelj^ saying that 
the volume is when the fourth point {x, y, z) is in the plane 
of the other three. 

The same six-fold volume can be expressed as the product of 
the double base P1P2P3 by the J_ 79 from P on the plane of the 
base. This p is found by bringing the Eq. of the plane into 
the normal form, by multiplying the determinant by F, where 



TRIANGLES IN SPACE. 241 



where x', y\ z^ are co-factors of a?, ?/, z in the determinant. 
Hence, the radical ->/J \ is the double area of the A P^P^P^. 
A\^e note that cc'-xK Z/'*^!? ^''^1 are the projections of PiPgPg 
on the Cd. planes H to X, Y^ Z., i.e., they are the Cds. of the 
A ; also, the negative terms are their (so-called) inner products 
in sets of two ; hence, 

The squared area of a A is the sum of its squared Cds. plus 
twice the sum of their inner products, in sets of two ; or 

The squared area of a A is the inner squared sum of its Cds. 

The Cds. of the tract P1P2 being the differences of the Cds. 
of its ends, by observing signs, the square of the tract may be 
expressed as above. 

15. The direction-'^s a, ^, y of a _L on a plane are called the 
Position-^s of the plane ; clearly they are also the diedral ^s 
of the plane with the Cd. ]}lanes^ since the ^ between two 
planes equals the ^ between _Ls on them ; they are the comple- 
ments of the ^s between the plane and the Cd. axes. 

The position-cosines of a plane are I : y/P + jn^ + n^, and 
two like ones. 

15*. In case of oblique axes, the position-cosines are still jF7, Fm^ 
Fn, but the position-Ys no longer equal the diedral 'Ys. But any diedral 
X, as between the plane and XY, equals the V between the ± jo on the 
plane and the _L p^ on XY. The direction-cosines of p are a, B, 7, or Fl, 

Fm, Fn ; those of pz are 0, 0, 5 : &j (see Art. 4*) ; put these values in (C2*): 
Fl, Fm, Fn for a, /8, 7, and 0, 0, 5 : w for aj, &^, 7^, and get 

pp^=p_Pz=.{c^\''Fn-x"Fm-y\>"Fl}:S.w\ 

= F{n{l-c^^)-m{x-fc^)-l{^\^-(^x)]-S.(^\. 
The cosines of the other diedral Ys are got by permuting symbols.- 

16. The ^ between two planes ttj, tts, whose Eqs. are 
liX 4- m^y + Tii^: -f c?i = 0, I2X -\- r)i2y -f- 712Z + (^2 = 0, 

equals the ^ between the _Ls on the planes ; hence 



242 CO-OEDINATE GEOMETKY. 

^-\ ___ — _ ' 

7ri7r2= (Zi?2+Wim2+ni?i2) :V/i^+mi^+ni^«V?2^H-m2^+n2^. 

The slope of a E.L. to a plane is the complement of the slope 
of the RL. to the J_ on the plane ; hence, if the Eq. of the plane 

be 

Ix + my -\-nz-\- d = 0^ 

and those of the RL. be 

x-xy ^ y — yi ^ z — z^ 

X [X V 



7rX| = QX + m^ + nv) : V/^+ w^+ n^-^X'^ /x^H- v^. 
Hence, the two planes are _L when 

lj,2 + 'yt^i'nfii + n^n^ — O. 
They are H when 

1^2 + Wjmg + ^1^2 = V?i^+ mi^+ ^1^ • Vy + m2^+ ^2^, 
i.e., when 

{l^m2 — km^y + (mi7i2 — m2ni)2 + (%Z2 — ^2^1)^ = 0. 

This sum of squares is only when each is 0; i.e., only 
when 

1-^^:12= mi : m2 = ni : ?22« 

This condition is, indeed, geometrically evident, since it 
declares only that the position-cosines of the two planes are the 
same. 

The plane and the RL. are H when 

IX + "mil + wv = 0. 
They are _L when 

zx + m/i + ?2v = VF+wF4^ VanvTv" ; 

i.e., when 

? : A = m : ft = w : T/. 

Hence Ix + my -\-nz -{- d = 

T X — X. y — 1/-, z — Zi , 

and = ~ = are ±,. 

I m n 



INTERSECTIONS OE PLANES. 243 

17. To find where a line meets a surface^ replace two of the 
Cds. iu the Eq. of the surface by their values in terms of the 
thirds taken from the Eqs. of the line ; thus is got one Eq. in 
one Cd., whose roots are Cds. of the points of meeting. The 
number of these roots cannot be greater than the number of 
common points, though it may be less^ since to any value of z 
(say) may correspond several values of x and y. If the Eq. in 
the third Cd. reduces to = 0, i.e., is satisfied for every 
value of that Cd., then^ and only then, the line has all of its 
points on the surface ; i.e., the line is on the surface. 

In the special case of the plane Ix + "iny -{- 72Z -\- d = and 
the EL. y = tz -^ &, x = uz + a we get 

{lit + 'I'^it + ^0^ + Q^^ + ^'"^^ -{- d) = 0, 
and this reduces to = 0, is satisfied for every z, only when 

lu + mt + n = O and la + tnh + c? = O, 
which Eqs. say the RL. lies in the plame. 

18. The common point of three planes, 

7iX+7?iiy + ??i2; + c?i = 0, 
l.jpc •+ rn^y + noZ -j- c?2 = 0, 
Lx + m.^ + ??32 -f- c/g = 0, 

is found hy solving the three Eqs. as simzdtaneous ; the results 

are 

x = — \d1m2 n2\:\Iim2 n.s], 

y = — \h ^h ^^3 \''\h '^2 '^h 1 7 

Z = — I ?i 7710 C?3| : I Ij 7712 713]. 

Hence, if | ?i 7719 713 1 = 0, the common point is at 00, the in- 
tersections are |1 RLs. ; if, besides, a numerator, as \lim2dg\, 
be 0, the common point becomes indefinite, the intersections fall 
together, the three planes pass through the same RL. 

19. If four planes meet in a point, their four Eqs. are satis- 
fied by the same triplet of values x, y, z ; this can be luhen, and 
onlyiohen, \i^m,n,d,\^0. 



244 CO-ORDINATE GEOMETRY, 

Or we may reason otherwise, thus: If 7ri = 0, 7r2 = 0, 
erg = be three planes, then Aitti + ^^2^2 + A, TTg = is a plane 
through their common point; for this Eq. represents a plane, 
being of first degree in x^ y^ z\ and it is satisfied by that triplet 
of values cc, 2/, 2, that satisfies the three at once. 

Four planes tti — 0, tts = 0, vrg = 0, 7r4 = meet in a 
point when four multipliers Ai, A2, A3, A4, can be found such 
that AiTTi + AgTTs + AgTTg + A47r4 = identically ; for the triplet 
of values x^ y^ 2, which reduces any three of the tt's to 0, must 
reduce the fourth tt to also. 

20. If ^1 = 0, N.2=0, be normal Eqs. of two planes, 
then Ni — A^2 = ^ is a plane through their common RL. ; for 
any triplet x, y^ z^ satisfying the first two Eqs., satisfies the 
third. Also, A = JVi : iVg 5 i-e., A is the ratio of the distances 
of any point of the third plane from the two base-planes, or A is 
the ratio of the sines of the slopes of the third plane to the base- 
planes. Hence, JVi — iV2 = resp. JV"i + -A^2=0 is the 
inyier resp. outer halver of the ^s between the base-planes. 

21. To find the direction-cosines of a RL. halving the ^ 
between two RLs directed by a, /?, y and a', ^', y', take two lis 
to the RLs., through the origin, and on each take a point dis- 
tant 2 from the origin ; the Cds. of the points will be 2 a, 2 (3, 

2y and 2 a', 2/3', 2y'; the mid-point of the two will be (a -fa', 

p-{- (3\ y -f- y') , and will be on the halver sought ; hence 

X _ y __ z 



a-fa' ^+^' y + y' 

is the Eq. of the halver; hence a + a', y8-f-/3', y + y'? each 
divided by V(a + a'y+ {(3 + B'Y + (y -f y')^ are the direction- 
cosines sought. The radical ^ reduces to 

V2-f 2(^ = 2 . ^ 

- 2 

where <^ is the ^ betiveen the RLs. 



SYSTEMS OF PLANES. 245 

To find the direction-cosines of the outer halver, it suffices to 
change the signs of a', ^', y' ; the radical -yj then becomes 



V2^2^ = 2.gj. 



22. A system of planes through a point may be called a pen- 
cil of planes; for a system of planes through a RL. no fitting 
name has yet been used in English ; perhaps duster would 
answer best to the German Bueschel^ suggested by the phrase 
in Architecture, clustered column. 

The common RL. of a cluster may be called its axis. Any 
two planes may be taken as base-planes of a cluster. 

To find what plane of a cluster 

l-^x -\- m{y + UiZ -{-di — \ (l^x + m^y -f n^z -f (^2) = 
is J_ to Zx 4- i^y +'nz + d = 0, we have at once, by Art. 16, 
IQi — XI2) -j-m{mi — Xm^ -f nin-^ — Xn.^ = 0, 

whence X is to be found, and, on substitution, the Eq. of the 
plane is found to be 

(II2 + mm2 -\- nn^ {l^x + m-^y + n-^z -\- d^ 

— (Ill + ^''^^1 + ^^^1) (h^ + ^^^22/ + ^^2^ + (^2) = 0. 
The position-cosines of this plane are proportional to 
m (lim2 — hi^ii) — n {n^2 — '^2^1) 1 
and two like expressions. 

But ?, m, 71, are proportional to the position-cosines of the 
given plane, say £Ca + 2//^ + ^y — i> = 0, and the paren- 
theses are proportional to the direction-cosines of the axis, say 

~ ' = '^ ~ ^ = ~ ; hence the above Eq. of the sought 
a (3' y' 

plane may also be written 

{X - x') (^y'-^V) + (2/ - y') (r« - y'«) 



246 CO-OUDINATE GEOMETRY. 

23. To find the plmie tlirough either of two RLs. H to the 
other, regard the RLs. as the axes of the clusters IIi— AIT2 = 0, 
Ilg — kU^ = 0. The two II planes of these clusters are the 
planes souglit. 

By Art. 16 

-■' = — i ^ = — = r (say) . 

^3 — K?4 m^ — Kin^ n^ — kii^ 

Clear each of the three Eqs. and solve for X ; so we get 

X = I ?j m3 7^4 I : I ?2 in^ ^4 1 5 

whence IIi 1 12 mg n^ | — Ilg | Zj 'm^ 714 1 = , 
and advancing the subscripts by 2, 

ITg I ?4 mi 7^2 1 — n4 Ks ^1 ^2 1 = 0) 
are the planes sought. 

Like results are reached by this reflection : The common _L to 
the two RLs., say 

X — x' _y — y' _z — z' , x — x" _y — y" __z —z" 

is clearly _L to the plane having the direction of both, |1 to 
both or through either |1 to the other ; the direction-cosines of 
this _L have already been found proportional to f3'y"— (3"y', etc. ; 
hence the planes are 

(x-x') (§'y" -§":/') + (y-y') (yV'-y'V) 

+ (z-z')(a'§"-a"§') = 0, 

and (x-x"){(3'y"-§"y')-{-"' 

The distance between these planes from any point, as (x\ y\ z') , 
of the first to the second, is plainly the (shortest) distance 
between the RLs. ; the same is got by reducing the Eqs. to the 
normal form, dividing by the second root of the sum of the 
squared coefficients of x, y, 2, and then putting x', y\ z' for x^ 
y^ z in the second Eq. ; now that sum is the squared sine of the 
^ between the RLs. (Exercise, Art. 6) ; hence, calling the 
distance d and the ^ ^, we have 



MOMENT OF TAYO IIIGHT LINES. 



24^ 



d . <j^l = (x'-x") (§'y"-§y) + (y'-y") (7'a"-:/"a') 

+ (z'-z"){a'§"-a"§'). 

This expression, called by Cayle}' the moment of the two 
RLs., may be written as the difference of two determinants, 
thus : 



d'^\ 



x' a' 


a" 


— 


y' §' 


§" 




Z- y' 


y" 





a;" a' a" 

.v" §' §" 

Z" ~y' y" 



which clearly 

= - la'{§"z"-y"y") +^'(y"x"-a"z") +y' (a"y"- I3"x") 

+ a''(§'z'-y'y')+/3\yy-a'z')+y''(a'y'-§'x')l. 

Now the Greek letters are the direction-cosines of the two 
RLs., and are but special values of A', /x', v' and X", /x", v" ; 
while the parentheses are what, consistently with Art. 11, must 
be denoted by A", M", N" and A', M', N' ; hence, disregarding 

sign, 

d-ct>\= X'A"+^'M"+v'N" + A"A'+/x"M'+v"N', 

which expresses tJie moment of tivo RLs. through their Cds. 

24. To find the volume of a tet^^aeder fi:Kedhy four planes, 
ni=0, etc., it suffices to repeat, step by step, the reasoning 
in Plane Geometry as to the area of a A fixed by three RLs. 

The result is quite of like form : 

6 T= \lim2n2dA^ :\l2msnA'\limQn4^\-\lim2nA'\lim2nJ. 



25. It is easy now to iuterpret A = (3z'—yy', M and N. 



Let 



Jb ^"^ aAj 



y —y' _z — z' 



be the RL. , then — - = -^ = — - = S 

§ 1 ^ § y 

is the Eq. of the RL. through the origin and (x', y', z') ; 
hence x'=a'S, ij'=l3'S, z'=y'8, and A = S{(3y'- y§'), 

M=8(ya'— y'a), N = S(a/5'- a'^) . The multipliers of 8 are 

the position-cosines of the plane through the two RLs. ; i.e., of 
the plane through the given RL. and the origin ; or, they are the 



248 CO-ORDINATE GEOMETRY. 

direction-cosines of the EL. through the origin _L to this plane, 
and hence A, M, N are the Cds. of a point on this RL. distant 
8 from the origin ; i.e., the Cds. of a point on this RL. as far 
from the origin as («', 2/', ^') is. 

EXERCISES. 



1. The tract P^P^ i^ ^^^ ^* ^' ^^ ratio n^:n^, P'P^ is cut at P" in ratio 



nj^ + n^: n^, P"P^ at P" in ratio n-,^-}- n^-h n^ : n^; find the Cds. of P'", the 
centre of proportional distances of P^, Pg, P3, P^. 

2. Eind distance of {x, y, z) from a RL. through directed by a, j8, 7. 

3. If III — ^} 112 = Oj ^3 = 0, n^ = do not meet in a point, the 
Eq. of any plane is of the form X{n.-^ + K.^Tl^ + A3n3 + A^n^ = 0. 

Show that the n's, regarded as Cds. of a point, are proportional to fixed 
multiples of its distances from the planes ; the A's, regarded as Cds. of a 
plane, to fixed multiples of its distances from the points A^ = 0, etc. 

4. Eind Eq. of : a plane through two II RLs., a RL. through two points. 

5. Show that the Eq. of a sphere of radius r, centre [x^, y-^, z-^) is 



X — x^ + y — yi +2 — ^1 =r\ 

6. "What, then, are the Eqs. of a circle in space ? 

7. Eind the centres of the inscribed and circumscribed circles of a A 
whose vertices are on the rectang. axes. 

8. Show that the three median planes of a trieder (through the edges 
halving the counter-szc?es or face-Ys) meet in a RL. 

9. Three planes through the three edges of a trieder meet in a RL. ; 
show that the compound ratio of the sines of the segments into which they 
cut the counter-sides is 1 ; and conversely. 

10. Any plane through the vertex of a trieder cuts the sides into seg- 
ments the compound ratio of whose sines is — 1 ; and conversely. 

11. The 6 planes of intersection of 4 spheres meet in a point. 

12. Three positive rectang. axes pierce a sphere about at X, Y, Z; 
X' is the pole of the circle through X, Y, Z; the point Xis carried up to 
X' along the great circle arc XX' ; find how Y and Z move, and the 
formulae of transformation from axes OX, OY, OZ to OX', OY', OZ'. 






SUEFACES OP SECOND DEGEEE. 249 



CHAPTER II. 

SURFACES OP SECOND DEGREE. 

(Quadrics or Conicoids.) 

26. The general Eq. of second degree has the form 
7coc^-^2hxy-{-jy^-{-2gzx-\-2fyz-\- iz'-\- 2 lx-{-2my-^2 nz-\-d — 0. 

It shall be referred to as F{x^ y, z ; x^ ?/, z) = 0, or Q — O, 
Before discussing it, certain general notions shall be premised. 

A surface may be thought as the locus of 2^ point ; it may also 
be thought as the locus of a Zme, or traced by a 'moving line. 

Suppose two surfaces F{x^ ?/, z\ ][)) = 0, ^(ic, y^ z\ p) — 0, 
contain the same parameter p ; for any special value of ^ they 
fix a single line as their intersection, while various values of p 
3ield various such lines ; by elimiuating p between the Eqs. we 
get a relation holding between x, y^ z for all points on all such 
lines; i.e., we get the Eq. of the locus of the line^ the surface 
traced by it moving. If F and </> contain two parameters p and 
p\ they must be bound together by some Eq., as /{p^p') = ; 
the number of such Eqs. of condition is, in general, one less 
than the number of parameters. 

The moving line is called the generatrix (in any one position 
it is an element) of the surface. The motion of the generatrix 
is commonly defined as gliding along fixed lines called direc- 
trices. Since the Eqs. of generatrix and a directrix hold for 
the same triplet x, y^ z, b}' eliminating these from the four Eqs. 
is got one Eq. of condition between the parameters for eacJi 
directrix; hence, when the Eq. of generatrix contains n param- 
eters, it must glide on n — 1 directrices. 

When the generatrix is a RL.^ the surface is called ruled. 
The Eq. of the RL. contains four parameters, hence three 



250 CO-OEDINATE GEOMETRY. 

directrices define its motion. Ruled surfaces that can be un- 
wrapped upon a plane are called developable ; the generatrix of 
such a surface always touches a fixed curve called cuspidal edge. 
Other ruled surfaces are called warped or twisted. 

27. A cylindric surface is the path of a RL. pushed. 

Be y = tz-\-b^ X = uz-\- a the E.L. ; then, since the direc- 
tion of the RL. changes not, t and u are constant ; let c^(a, h) =0 
be the Eq. between the parameters a and 6, yielded by the Eq. 
of the directrix ; on replacing a and h hy their values results 

(^ {x — UZ.I y — tz) = 
as Eq. of the cylinder. 

Or, be Ix -{-my -\-7iz-\- d=p, l'x-\- m'y -{-n'z -\-d'= p' the 
Eqs. of the RL. ; letting only p smdp' vary, we keep each plane 
II to itself, and hence all the intersections || ; if <^(i>,p') = 
be the Eq. between the parameters, the Eq. of the cylinder is 

cl)(lx-}-my-j-nz-\-d, l'x-{-m'y-\-n'z-\-d') = 0. 

Hence any Eq. of a cylinder is an Eq. between two functions 
of first degree in x, ?/, z; and the converse is clear. 

28. A conic surface is the p)Gith of a RL. turned (about a 
point) . 

X — Xi y — yi 
Be ; =p, '— = q the RL., {x^, y^, z^) being the 

fixed point about which it turns ; then, just as above, 

/x — Xi y — vA 
^\z -z^ z -zj 

is the Eq. of the conic surface (or cone). We note the Eq. is 
homogeneous in Cd. differences ; the converse is clear, that every 
Eq. homogeneous in Cd. differences pictures a cone. 

If the fixed point, or vertex of the cone, be the origin, the 
Eq. is homogeneous in x, y, z; and conversely. 

Both cylinder and cone are developable ; the cuspidcd edge of 
the cone is reduced to a point, the vertex, while the cylinder is 
but a cone with its vertex at oo. 



DISCEIMINANT OF THE QUADEIC. 251 

29. A surface of revolution is the path of a line revolved 

about a fixed JRL., the axis, to which it is supposed rigidly 
attached. Or, it is the path of a circle whose (varying) diam- 
eter is always halved by a fixed RL., the axis^ at right angles. 

The generating circle in any position is called a parallel, the 
revolving line in any position, a meridian, of the surface. 

If .^ = ^ — ^ = i he the axis, then lx-\-niy-{-nz=p 

I m n 



is a plane _L to it, and x — Xi -\-y — yi -\-z — Zi = ^'^ is a 
sphere about Xi, yi, Zi ; an}^ parallel is the intersection of two of 
these surfaces ; hence, exactly as before, 



^{x — x-^ -\- y — y{ -{- z — Zi , lx-{- my-\- nz) = 
is the general ^q. of a surface of revolution. 

30. Returning to the Eq. Q = 0, we note it may be 
written 

(kx+hy-\-gz-\-l)x+ {hx-{-jy-{-fz + m)y 

-{-{gx-\-fy-\-iz-{-n)z-\-{lx-\-my-{-nz-\-d) = 0. (1) 

The parentheses may be called, in order, Q^, Q^, Q^, Qi. 

Again, we note there are ten coefficients ; but, by dividing 
by any one, the number is reduced to nine ; these may be deter- 
mined by nine independent Eqs. ; hence, nine simple conditions 
are needed and enough to determine a quadric. 

To pass to II axes through a new origin, x', y\ z\ put x-\-x\ 
y-^-y\ z-i-z' for x, y, z; then, by reasoning quite like that in 
Plane Geometry, the result is seen to be 

kx^ + 2 hxy -f jy^ + 2 gzx + 2fyz + iz^ 

+ 2QJ-x+2QJ-y + 2QJz+Q'=0, (2) 

where Qx'= kx' -{- hy' -\- gz' -\- 1, and so for the others. 

We note that the coefficients of terms of second degree 
change not. 

TVJien, and only when, all the Q's vanish, does the Eq. become 
homogeneous in x, y, z; but then it represents a cone through 



252% 



CO-OHDiNATE GEOMETEY. 



the new origin x\y',z'. Now Q' = Q':,'X' + Q\-y' + Q','z' -\-Q\\ 
hence, for Q'^., Q'^,, Q'^, Q' each to vanish is the same as for 
Q'x-) Q'yi Q'zt Q'l each to vanish ; and these four vanish for the 
same triplet x', ?/', z' when, and only when, 



A = 



k 


h 


9 


I 


h 


J 


f 


m 


9 


f 


i 


n 


I 


m 


n 


d 



= 0. 



Hence, Q=0 represents a cone wlien, and only when, 
A = O, This A may be called the discriminant of Q = 0. 

If A = , then /cx^ + 2 hxy -\-jy^ + 2 gzx + 2fyz + iz^ = , 
and this breaks up into two linear factors in x, y, z] i.e., the 
cone breaks up into ttvo planes, when, and only when, 



D = 



k 


h g 


h 


J f 


9 


f i 



= 0, 



as was proved in Plane Geometry. Hence Q = represents 
two planes when, and only when, A = 0, X) = 0. 

31. If Q'. = 0, Q', = 0, Q'.= 0, but Q'(orQ'0<0, 
then if any triplet {x, y, z) satisfies the new Eq., so does the 
counter -triplet (— x, —y, —z), since the Cds. appear only in 
pairs; i.e, if any point be on the surface, so is its counter-point 
as to the new origin; i.e., the new origin halves every chord 
through it, and is the centre. 

The Cds. of this centre, or x' , y', z', are found from the three 
Eqs., Q', = 0, Q\ = 0, Q', = to be L:D, M-.D, 
N:D. Hence, if X>^0, the centre is ^?^;^?^^Y^/, and the sur- 
face is centric ; if X) = 0, the centre is at oo , the surface is 
called non-centric. 

One or more of the numerators L, M, N ma}' vanish along 
with D ; the centre is then indeterminate. 

In case Q' alone = 0, the origin (x\ y\ z') is on the sur- 
face. Call the sum of the six terms of second degree S ; then 
the Eq. becomes 



TANGENTS. 253 

s-^2{xq\-\-yq\-^zq^,) = o (3) 

Draw through the origin any RL. x : a = y : ^ = z : y = p. 

To find where it meets the surface, replace «, y, z in (2) ; hence 

V + 2(_aQ', + W.4-7Qyp = 0.* (4) 

One root pi of this Eq. is always ; the other, p2<, is also 
when, and only when, 

or when x- Q', + y • Q',-{- z- Q', = ; (5) 

i.e., the ML. meets the surface only at the origin, and then in 
two consecutive points, when, and only when, it lies in the plane 
whose Eq. is (5). 

A RL. meeting a surface in two consecutive points is tangent 
to the surface ; the plane containing all tangents to a surface at 
a point is tangent to the surface at that point. Hence (5) is 
the plane tangent at the origin. 

To find the Eq. of this plane tangent at (x', y', z') in terms 
of the old Cds., replace x, y, z hj x — x', y — y', z — z'; hence, 

{X - x') Q'^ + (2/ - y') q\ + {z- z') q', = 0. (6) 

Add x'q', + y'q.', + z'q'^ + q\ = o, 

since (x', y', z') is on the surface ; hence, 

= x'Q, + x'q, + z'q^+q,; (7) 

or, F(x,y,z',x',y',z') = 0=F{x',y',z';x,y,z), (7*) 

is the Eq. of the plane tangent to F(x, y,z; x, y,z) = at 
(x',y',z'). 

32. The meaning of the fact that this Eq. is like-formed as 

to X, y, z and x', y', z' is quite like the meaning of the like fact 
in Plane Geometry, and is developed in like wa3^ In fact, if 
{x', y\ z') be not on the surface, the Eq. still represents a plane, 

* 2 is what S becomes on putting a, $, y for x, y, z. 



254 CO-OBDINATE GEOMETEY. 

being of first degree ; also, if a plane through (cc', y\ z') touches 
Q = at (a^i, 2/i5 ^i) 5 its Eq. is F{x^, y^, z^^-, x, y, z) = 0, and 
hence F{xi^ y^, % ; x'^ y', z') = ; but this Eq. also says that 
(.Tj, ?/i, Zj) is on F(x^ ?/, z ; a;', y', z') = : hence this last Eq. 
is that of a plane through all points of tangency of planes 
through {x', y', z')- Such a plane is called the polar (plane) 
of (o)', y', z') as to Q = 0. 

33. Hence it appears that all tangent-planes, and hence all 
tangent-lines, through a point, touch a quadric (surface of sec- 
ond degree) along ix, plane-section of that surface. But ?^ plane- 
section is clearly a conic; for the section made by the Xy-plane 
is found, by putting 2; — 0, to be the conic 

ka? + 2 hxy -\-jy^ + 2 Ix -f 2 niy-{- cl = 0, 

and any plane may be taken as Xl^-plane without changing the 
degree of the Eq. or its general form. Hence all tangents 
through a point, or the tangent-cone through a point, touch the 
quadric along a conic. 

We ma}^ note in passing tliat || p)lane-sectious of a quadric 
are similar conies. For the}^ are got by giving different con- 
stant values to 2:, as c, c', etc ; but these do not affect the Jirst 
three terms of the conic, on whose coefficients alone, k, h,j, the 
shape of the conic depends. 

34. The icliole theory of poles and polars, since it depends 
solely on the symmetry of the Eq. of the tangent-RL. resp.- 
plane as to the current Cds. and Cds. of the pole, may now be 
repeated from Plane Geometiy. 

Poles lying each on the polar of the other are conjugate. 

Planes each through the pole of the other are conjugate. 

Tangent-planes along a conic on a quadric go tln'ough a 
point. 

Poles of planes through a point lie on a plane. 

As a pole moves about on a plane, its polar-plane turns 
about a point ; as a plane turns about a point, its pole moves 



COXJUGATES. 255 

about in a plane. A tract from a pole to its polar-plane is cut 

harmonically by the quadric {referee) . 

35. If (i : D, J/: D, X: D) be taken as pole, then Q',, Q\, 
Q\ vanish, and the polar is Ox + Oy + O2; + Q'l = ; i.e. , tiie 
polar is the plane at x. Hence, the polars of all points at x 
pass through this point, the centre. All ELs. through the same 
point at CO are || ; this point being the outer mid-point of all 
intercepts (chords) of the quadric on these RLs., their inner 
mid-points lie on the polar of the point at 00 ; this central plane 
accordingly halves all \\ chords through its pole (at x) ; i.e., 
halves all its conjugate chords. Hence it is called a diametral 
plane. Among all these chords is one centred one, \yhich is 
therefore a diameter conjugate to the diametral plane. The 
section of the diametral plane, being a conic, itself has an x of 
pairs of conjugate diameters; any one of these forms with the 
common conjugate diameter a triplet of conjugate diameters; 
the three planes fixed by the triplet of conjugate diameters form 
a triplet of conjugate diametral planes. Each plane halves all 
chords II to the intersection of the other two. 

The poles of a system of |1 planes lie od the diameter conju- 
gate to the planes. The centred distances of a pole and its polar, 
measured on the diameter thi'ough the pole (conjugate to the 
polar) , have for their geometric mean the half of that conjugate 
diameter. Tangent planes at the ends of a diameter are \\ to 
its conjugate diametral plane. 

36. The notion of diameti'al plane ma}' be got otherwise, 
thus : 

Be o^-c^ ^y-y^ ^z-^ ^ ^ 

a ^ y 

or X=x'-^ap, y = ?j'-{-pp, z = z'-\-yp 

a RL. ; combining with the Eq. of the quadi'ic, we get 

V + 2Tp + Y = 0; (1) 

when 1 has its former meaning. 



256 CO-ORDINATE GEOMETEY. 

and, lastly, 

Eq. (1) has two roots, pi, p2 ; i-e., every RL. meets a quadric 
in two, and ojily two, points. These roots are counter when, 
and only when, T = ; i.e., for a, ^, y held constant (for 
II RLs.), the intercept or chord of the quadric is halved by 
(x', 2/', 2') so long as the point {x\ y^, z') lies in the plane, T— ; 
this latter is the Eq. of a plane, being linear in x\ y', z'. 

From the Eq. of this diametral plane, 

it is seen that all diametral planes form a pencil through the 
intersection of Q\ = 0, Q\j = 0, Q'^ = ; i.e., the centre. 

37. To find the mutual slope of conjugate plane and chords, 
put 0-, T, V, ^ (read koppa) for ha + hf^ + gy-, ha 4-j/3 +/y, 

ga +//5 + 2^V? ^^ + ^^/^ + ^y 5 t^®" i^ 
T = o-x' -f T?/' + 1;;^' + 7 = 0, 
and the position-cosines a', /3', y' of this diametral plane are 
a' = (r:E, §' = r:E, y' = v : H, where E' = cr" + r^ -j- v' . 

Hence, if ^ be the slope in question, 

<f>\ = aa'-{-§§' + yy' = ((Ta-\-T^ + vy) : E=%'.E, 

The important question arises : Are conjugates ever perpen- 
dicular? If so, <j?) = 90°, </)|=l, % = E, a = a', ^ = ^', 
y = y' ; whence, 

a = cr:E, p = r:E, y=v:E; (1) 

or, (era -\- t(3 -\- vy) a = O", 

(o-a -f T^ + vy)(S = r, 
(era -f- Ty8 -j- vy) y = v. 



PERPENDICULAH CONJUGATES. 257 

Here then are the three Eqs. to determine the a, /8, y of a 
chord _L to its conjugate diametral playie; there is a fourth Eq. 
connecting them, a^ + /3^ + y^ = 1, but this imposes no new 

fourth condition, since it can be got from the three by mul- 
tiplying by a, j3, y in turn, adding and cancelling. Actually 

to find a, (3, y from these four Eqs. would be very tedious ; 
it is better to determine i? or :S, and thence a, /3, y. On 
replacing o-, r, v in (1) by their values there result 

7ia+(j-R)l3+fy=0, 

ga+fl3 + (i-B)y =0. 

Divide in turn by %, /A, gf; add in turn a :/, /3 : g, y : h; 
put A, B, CfoY 

jc-% j-f±^ ^-^; 

f g h 

also put U' for a:f-\-fi:g-^y:h\ 

hence result U= a{B — A) : hg, 
, U=§{B-B):fh, 

U==y(B-C):gf; 
or, a = hgU: (B — A), 

P=fhU:(B-B), 

y = gfU:(B-C). 

Squaring, adding, and re-membering a^ -\- (3^ -\- y^ — I ^ we 

get 

1: jj=-^\hY iB-A' +fh^ '.B-:^ + gY-'.B- C^\ ; (2) 

and on dividing in turn by/, ^, 7i, and adding, we get 

^Jig U fh U gf U . 
f' B-A'^ g ' B-B^h ' B-C' 



258 CO-ORDINATE GEOMETRY. 



whence 1 = -^ 

/ 



or U=^ 0. 

On multiplying by 
K = 1 : ligf^ results 

{R -A){R- B) {R - G) 
1 1 



A'^ g ' R-B ' A ■ B-C' 






(3) 


{B-A){R-B){R-0), 


and by 



f\R - A) g\R - B) h\R - C) 



= 0. 



This Eq. is of third degree in R^ hence has at least one real 

root. To decide about the other roots, suppose k + and 

A<B<C; then, calliag the left side of the Eq. j&, we see 

that 

for i2 = — 00, ^ is — ; for i? = ^, ^ is — ; 

tor R = B, jE; is + ; for R=C, E is - ; 

for i?= -f CO, ^ is +. 

In case k is — , a change of sign in E takes place between 
R=z — Qc and R=^A^ instead of between R= C and R= oo. 
In any case E changes sign thrice as R passes through real 
values from — oo to +go ; i.e., in any case the Eq. E = 
has three real roots : Ri, R2, R^. 

These real values of R give three real triplets of values of 
a, /5, y ; hence, there are, in general., three, and only three, 
diametral planes _L to their conjugate chords. They are called 
chief (or principal) planes of the quadric. 

38. Each of these roots R^, R^., R^ must satisfy Eq. (3) ; 
hence result three Eqs. ; take second from first, and multiply 
by hgf\ hence, 

^ ^ , W ^ f^i' 



-^A{R.- 



A){R,-A) {R^-B){R,-B) 
'^ {R,-C){R,-C) 



= 0. 



PKINCIPAL PLANES. 259 



Since JSg — -^! cannot =0, the parenthesis J j must =0; 
and two like Eqs. are got by permuting the indices. 

Now ai = hgUi: {Ri — A)y 

and so for the indices 2 and 3. Calling the three directions of 

the diametral planes, or what is tantamount, of their conjugate 

chords, di, cZg, c?3, and dividing by the U's^ since they can 

/-^ /'~H ^~^ 

none of them = 0, we get dic/g = 0, dgC^g = 0, d^di = ; i.e., 

the three chief planes of a quadric are ± to each other. 

39. In the special case U=0, follow also Ii=A=B=C ; 
hence. 



'5 



k = A-h^^, <T = Aa + hgU, R = A + ligfU\ 

whence U-fU'a=0, U-gU'(3 = 0, U-hU'y=0. 
Hence a=l :fU, 13=1 :gU, y=l :7iU, or U=0. 

Squaring and adding the first three Eqs., we get 
U = ^\7iY-{-fh'+ff^ri:hgf. 

This is the same value of U as is given by Eq. (2) when 
A = B = C ', but besides the triplet a, ^, y thus got, the prob- 
lem is solved by any other triplet a', /5', y' that makes U= 

or makes - _|_^_}-^ = 0. This Eq. is satisfied in an oo of 

f 9 ^^ 
ways and whenever U(aa'-\- /S/S' + yy') = 0, since 1 :/= Ua, 

l:g=XJ(3^ l:h= Uy. 

Now U is here not = ; hence 

aa'+^^'4-7y'=0; 

i.e., every direction _L to the direction (a, 13, y) is a chief 
direction. 



260 CO-ORDINATE GEOMETRY. 

It is easy to prove analytical!}', but it is also clear geometri- 
cally, that this special case is the case of surfaces of revolution ; 
the direction (a, /5, y) is that of the axis. 

If Jc=j = i and h = g=f the surface is a sphere; 
every triplet (a, /3, y) fulfils the conditions, every direction is a 
chief one. 

40. Returning to the Eq. of a diametral plane T=0, on 
putting for the Q's their values, it becomes clear that a triplet 
of conjugate diametrals are H to the Cd. planes when, and only 
when, h = 0, g = 0, f= 0, the reasoning being quite like 
the corresponding in Plane Geometry. Hence, by choosing as 
Cd. planes three planes II to a set of conjugates, we make the 
terms in xy, yz, zx vanish. This can always be done. 

Again, by choosing the centre as origin, we make the terms 
in ic, ?/, z vanish. This can be done only when the centre is in 
finity. But when the centre is at oo, the origin can be taken on 
the surface, making the absolute vanish, and also the term in 
z^.) SL diameter being taken as ^-axis. Hence the forms reduce 
to 

kx^ -\-jy^ + iz^ = d and kx^ -\-jy^ = 2 nz. 

When d and n are not = 0, the varieties of these are 



a^ W" (? 0^ IP" a^ 



2 2 

and —±^ = 22;. 

a 

m 

The first is an ellipsoid: real resp. imaginary. 

The second is an liyperholoid: single resp. double. 

The third is 2i paraboloid : elliptic resp. hyperbolic. 

When d = or ti = 0, the Eq. becomes homogeneous, 
and so represents a cone^ and, in case another coefficient van- 
ishes, still more specially a cylinder. These limiting cases the 
student himself can readily trace out. 

The rectangular conjugate diametrals recommend themselves 



DISTANCE-PKODUCTS. 261 

as Cd. planes ; to indicate that oblique conjugates are chosen, 
it suffices, as in Plane Geometry, to accent the constants a, 
5, c. 

41. Returning to the Eq. V + 2T/3 + Y = 0, which fixes 
the distances /oi, p^ from {x\ y', z') in the direction (a, y3, y) to 
the quadric Q = 0, we note that % depends only on the direc- 
tion (a, ^, y), and Y only on the point {x', y', z') ; hence, 
taking two points P' and P", and one direction, we find the 
quotient of the distance-products pi p2 ' pi" P2" is independent 
of the direction ; or, taking one point and two directions, we find 
the like quotient is independent of the point. Hence : 

The rectangles of the segments of two intersecting chords 
are proportional to the squares of the |1 diameters. 

Tangents from any point to a quadric vary as the H diameters. 

The areas of sections conjugate to a diameter vary as the 
rectangles of the segments into which they cut the diameter. 

Proof is quite as in Plane Geometry. 

42. We have seen that the six terms of second degree are 
unchanged by a mere change of origin; to find what functions 
of the coefficients are unchanged by a change of axes, proceed as 
in Plane Geometr}^, thus : 

Let the coefficients A;, J, ..., i change into 7c', f, ..., i', and 
the sum S of the six terms change into jS' ; then S = S' ', also 

0^ -f- 2 (j)xy -j- 2/^ + 2 if/zx + 2 xV^ + ^^ 

= x"-+2My'-\- y"+2fz'x'+ 2x'y'z'-\- z'^, 

since each is the squared distance D^ from the common origin 
to the same point F(x, y, z) or P{x', y', z'). Hence jS+fxD^ 
is not changed by change of axes ; hence the values of fi which 
make S + fxD^ = are the same for all axes, and specially 
the values of jx that make S -\- fxD' resoluble into two factors of 
first degree in {x, y, z) are the same for all axes. This resolu- 
tion is possible when, and only ivhen, 



262 



CO-ORDINATE GEOMETRY, 



7i + /xo) i + /x f + f^X 



= 0, 



as was proved in Plane Geometry ; for the form of S-{-ijlD^=0 
is the same as that of F(x, y ', x, y) = 0^ as is seen on putting 
z = l. The roots of this cubic in /x, called discriminating cubic 
when (0 = i/^ = ;)( = 90°, are the same for all axes; hence, on 
making the coefficient of /x* 1, the other coefficients must be 
constant /o?' all axes. 

^2 



They are 



-2l{M-gf)^_ + {gj-fh)rj^+{fTc-gh)^-\]:S\ 
:S\ or DiSK 



7c 


h 


9 


h 


J 


f 


9 


f 


I 



It is to note that the binomials are all co-factors of elements 
of either S^ or D. 



Geometric Interpretation. 

43. 1. For rectangular axes, w, \f/, x vanish, to|, i/^|, x| become 

each = 1 , and 5^ becomes 1 ; hence Jc -j-j -\-i= constant. Sup- 
pose S = 1 the central Eq. of a quadric, then k, J, i are the 
squared reciprocals of the half -diameters ; therefore, the sum of 
the squared reciprocals of three rectangular diameters of a quadric 
is constant. 

2. If conjugate diameters be taken as Cd. axes, h^ ^, /vanish, 
and the constants are 

Qcj+ji+ik):S\ 
hji : S\ 



THE SPECIAL QUADllICS. 263 

On dividing the first by the third, we get ~-\ \- - = constant ; 

i j k 

i.e., the sum of three squared conjugate (half-) diameters of a 
quadric is constant. 

Extracting the second root of the third, and inverting, we get 
5 :^kji = constant ; i.e., the volume of the parcdlelepiped fixed 
by three conjugate (half-) diameters is constant ; hence, its eight- 
fold, the volume of the parallelepiped of three conjugate diame- 
ters, or bounded by six tangent planes at the ends of conjugate 
diameters, is constant. 

On dividing the second by the third, there results 

|2 I |2 |2 

— + — 4" ^ = constant; 
ij ki jk 

i.e., the sum of the squared parcdlelograms fixed hy three conju- 
gate (half-) diameters^ taken two by two, is constant. The axes 
of the quadric being 2 a, 2 6, 2c, the values of the above four 
constants are 

—--{ — --\ — - , a^ + 6^ + c^, abc, ah + 6c + gcC- 
a- h~ c^ 

The third constant is abc = a'b'c'S. Hence Sira'b'c'S is 
constant; or, since 5=a)|'^, (47rcc'5'a)|) (2 c'- ^) i^constan*. 
Here the first factor 47ra'6'aj| is the area of the central section 
of the plane XY\ and the second, 2c'-^, is the projection of 
conjugate diameter on the _L to that plane ; i.e., the first factor 
is the base of a cylinder touching the quadric along a central 
section, the bases themselves touching the quadric at the ends 
of the conjugate diameter ; while the second is^ the height of 
that cylinder. Such a cylinder may be said to be |1 to the 
diameter. Hence, the volum.e of a circumscribed cylinder \\ to a 
diameter of a quadric is a constant : Sirabc. 

The Special Quadrics. 

9 O O 

/v*-* qi- ^y- 

44. The quadric - — h-^H — - = — 1 has no real points, 

or b-' c- 

since the sum of the squares of no three reals can be —1. 



264 CO-ORDINATE GEOMETRY. 

The section of the Xl'-plane is got by putting z = ; it is 

the imaginary £ — + ^-^ = — 1 ; all 1| sections are also imag- 

inary £'s. The like may be said of sections |1 to the other Cd. 
planes. Hence the surface may be called Imagiiiary Ellipsoid, 
with axes 2ai, 2bi,^2d. 

9 

The sections of — 4- ^ -j- — = 1 made by the Cd. planes 
a- b'- C" 

2 2 2 ^2 

are the real £'s, ^^ + 1^=1, l,^-^^=l, ^_}_i = l. All 
cr ■ b- b- cr a^ c- 

II sections are similar £'s. Hence the surface may be called 
Real Ellipsoid, with axes 2rt, 2 6, 2c. We may suppose 
a'>b^G\ i.e., 2a the greatest^ 2c the least, 2b the mean, 
axis. For a? > a, or y>b, or 2;>c, the sections become 
Imaginary £'s ; hence the surface lies wholly in the parallele- 
piped whose edges are = and || to the three axes. 

45. The plane sections of the ellipsoid are in general eZZ^pses; 
are they ever circles? That they are, is made clear geometri- 
cally, thus : Pass a plane through the greatest and mean axes ; 

it cuts out the £ ^- 4- -^ = 1 . Turn the plane about the mean 
a- b-" 

axis, 2 5 or Y; the section remains an £ of which 2 6 is still 
the minor axis, but the major axis gets smaller; when the plane 

y2 ^2 

is turned through 90°, the section is the £ ^ -\ = 1. of 

0- c- 

which 2 6 is the major axis. At some stage 2 b must have ceased 
to be minor and become major; at that stage the axes of the £ 
were = , the £ was a circle. To find the slope of this cyclic 
plane to the greatest axis, we have the Eq. 

62 = c^:/l- ^'~^' . ^2 



whence 0\= c Va- — W:b V6^ — c^. 

Of course there are two cyclic central planes, one sloped 6, 
the other — 0, to the greatest axis. All planes H to these cut 



CYCLIC PLANES. 265 

the ellipsoid in circles, growing smaller as the cyclic gets farther 
from the centre, and vanishing in so-called umbilics, or, better, 
cyclic points, as the planes become tangents. Clearly there are 
four such pointb. 

When a = b, the ellipsoid becomes one of revolution, made 
by turning the £, whose axes are 2 a, 2 c, about the less axis 2 c ; 
the sections J_ to this less axis are all cncles clearly, the two 
series of C3'clic sections falling together in them. This ellipsoid 
is sometimes called an oblate spheroid. The earth-surface is 
nearly such an ellipsoid. 

"When b = c, the ellipsoid, called pi'olate spheroid, is formed 
b}' turning an £ about its greater axis 2 a. The two series of 
cychc planes fall together _L to the axis 2 a. 

46. An important way of looking at the ellipsoid is to look at 
it as a strained sphere. Suppose a sphere of radius a to have 
all its chords H (say) to Y'-axis shortened in the ratio b : a, 
and all H (say) to Z shortened in the ratio c-.a; then, if 
P'{x',y',z') be any point of the sphere, and OP' be directed 
by a, /5, y, we shall have 

ic' = aa, y' = 13a, z' = ya, 

and if P (x, y, z) be the corresponding point on the surface 
got by working on the sphere as stated, we shall have 

x = aa, y = ^b, z = yc; 

whence, on squaring and adding, results 

O 9 9 

the surface is an ellipsoid. We may call a, ^, y the eccentric 
^s of P or OP. 

Since, in the shortening prescribed, H and = tracts remain 
!l and = , it follows that conjugate planes and diameters in the 
sphere remain conjugate in the elU'psoid; but in the sphere con- 
jugates are _L ; hence conjugates in the ellipsoid correspond to 
J_s in the sphere. Hence, if 



266 CO-OEDIKATE GEOMETRY. 

\^ii ^25 €3;, V^ 1? e 2? c 3;, ^€ 1, € 2> e &) 
be eccentric ^s of three conjugates, 

fl^ li^^2^ 2T^^3€ 3 — ^? 

€ i€ 1 "h e 2^ 2 ^^ t 3^ 3 — ^7 

jtt It ijn II _i III II _ A 
e i€ 1 -r e 2^ 2 T^ ^ 3^ 3 — ^' 

The student may now show that the S2cm of the squared projec- 
tions of tlii^ee conjugate diameters on any RL. or plane is con- 
stant. 



47. The Eq. ±i + ^ 4-51 = 1 of the tangent-plane at 



^2 -t- ^2 -t- ^2 

(x', y\ z') becomes in the eccentric form -e'i+ ^e'2 +-€'3= 0. 

a- b- c- 

On squaring and adding the Eqs. of three tangent planes at the 
ends of three conjugate diameters, the locus of the intersection 

2 9 9 

of the three is found to be the ellipsoid h — + — = 3. 

a^ b^ (? 

48. The Normal Eq. of the tangent plane is 
ax-\- ^y + yz = p, 
where a, /5, y are position-cosines. Comparing, we see 

px' py^ pz' 

, 1 x'^ 2/" ^" 

and = 4- ^ _|_ • 

p^ a^ b^ c* 

/^I2 12 ^I2\ 

Hence aW + ^W + y'c' =p'^ + L^ 4- ^ ] =p\ 

\a^ b^ (? j 

Hence the Normal Eq. of the tangent planes is 



aX-h ^y + yZ —^a?d^ -f- (S^'b- + y^C^. 

On squaring and adding the Eqs. of three such planes mutu- 
ally J_, the locus of the intersection is found to be the director- 
sphere x^ + y'^ -\- z^ = a^ -\- b^ -\- (?. 



THE HYPERBOLOIDS. 267 



0(? if _ ^ 



49. The quadric -^ + ^ ^=1 is cut by the X y-plane 



in an £ 1--^ = 1, as is seen on putting z=^0. All 

II sections are similar £'s, only larger the farther from the XY- 

11- z^ 
plane. It is cut by YZ in the H ■— — ^ = 1. The \\ sec- 

tions are similar W^^ smaller the further from YZ^ till x=a^ 

if z^ 
when the section becomes a pair of RLs. ~ =:0. Thence 

the sections are secondary //'s, flattening out towards a pair of 
II RLs. as X nears ± co. Like remarks hold for sections |I to 
the XZ-plane. Hence this surface is called an Hyperboloid 
simple or of one sheet. 

By reasoning like that in case of the ellipsoid, it is shown 
that this hyperboloid is cut in circles by a central plane through 
the greatest axis and sloped 6 to the rtiean axis, where 



a' 



._,:(l_^%.), 



^1 = ± cVa^ - 62 : a^W + g\ 

All planes |I to these are themselves cyclic planes, cutting the 
surface in ever larger circles. Hence, the single hyperboloid 
has no cyclic points. 



50. The quadric — r + ^ = — 1 is cut hy XYm the 

a" h" C2 

imaginarj' £ 1- ^ = — 1 ; the |I sections remain imaginary 

a.2 6- 

till 2; = ± c ; thence the £'s are real and grow ever larger., 

with z nearing ± 00. The section of YZ is the secondary /^', 

y'i 2;- 

^ = — 1 and II sections are similar, with ever larger 

axes. Like remarks hold for sections H to XZ. Hence, this 
surface is called an Hyperboloid double or of two sheets. 

The student can readily convince himself that the cyclic planes 



268 CO-ORDINATE GEOMETRY. 

of the simple hyperboloid are also cyclic planes of the double; 
the ch'cular sections shrink into fou7' cyclic points as they retire 
from the centre. 

These two hyperboloids have clearly a common asymptotic 

9 9 9 

cone Y- = 0, which has common cyclic planes with 

a^ If r 

them. 

51. The clearest notion of these three surfaces is got thus : 
Turn an equiaxial H^ its conjugate H\ and their common asymp- 
totes around the conjugate axis ; the H will trace out an equi- 
axial simple hyperboloid of revolution, the //' an equiaxial 
double hyperboloid of revolution, the as3'mptotes the common 
equiaxial asymptotic cone of revolution. Now change (say 
shorten) all chords _L to ZX in the ratio h : a\ all chords _L to 
Xy in the ratio c: a\ the resulting surfaces will be the surfaces 
in question. 

The circle — - -f ^ = 1 traced by the vertex of the revolving 

a-^ a-' 

H is called circle of the gorge; the corresponding £ —- -f- ^ = 1 

a-' Ir 

is called ellipse of the gorge. 

52. Passing now to non-centric quadrics, we see that the first 

\- ^^ = 4:Z is cut by XI^ in the point (0, 0, 0) , the origin, 

a b 

while all |1 sections are £'s : real for ;s > 0, imaginary for z<,0. 
The section of YX is the P y^ = 4:bz, while that of ZX is 
the P a? = 4:az. The surface may be thought made by a vari- 
able £ moving always |1 to XY, with its vertices on these two 
P's. The surface is called Elliptic Paraboloid; 4 a and 46 are 
its parameters; is its vertex. Suppose the P y^ = 4:a.z to 
turn around its axis, the Z-axis ; the surface generated will be 

CC 11 

the paraboloid of revolution -—-4-^ = 42;. Now suppose all 

a-^ (T 

?/'s, or all chords _L to XZ, shortened in the ratio b\ a\ the 
surface got so is -^ -f- ^ = 4^. 



IMAGINARY CYCLICS. 269 

As to cyclic planes, we may reason thus : Turn a plane H to 
XZ about the major axis of its elliptic section ; the minor axis 
grows, and the £ tends to a P^ as the plane turns through 
90° ; at some ^ the minor must have become equal to the 
major axis, the £ must have passed over into a circle. The 
slope of the cyclic planes to XY^ is readily seen to be 6 when 
^ = ±V6:a, a>b. 

The student will readily see there are two cyclic points. 

53. The second non-centric ~==4:Z is likewise tano^ent 

a b "^ 

to XT^at 0, as is seen on writing out the Eq. of plane tangent 

at (0, 0, 0) : ^-^^ = 2(^+0), or ^ = 0, which is 
a b 

the Xy-plane. But XY cuts the surface along the pair of 

RLs. —=:^. All II sections are M's: primary for 2;>0, 
a b 

secondary for z<.0. The sections of YZ and ZX are the P's 

if- =. — 4:bz, x^ = AiCiZ. The surface may be thought made by 

an H moving, always |I to XY, with its vertices on one of 

these two /''s. In all positions the asymptotes of the H are 

II to the pair _ — ^i-. As the H nears the Xy-plane, it 
a b 

passes over into this pair of RLs., then into its conjugate, 

w^hile its vertices pass over from one P on to the other. To 

two cowiier-values, -\-z, —z, correspond two conjugate //'s. 

The surface is named Hyperbolic Paraboloid ; 4 a and 4 b are 

its parameters, is its vertex, Z is its axis; it is saddle-like 

in shape. (See Figs, at end.) 

54. It is easy to see geometrically that the cyclic planes thus 
far determined are all the real ones. For the diametral plane 
of the chords of the circles must be _L to them ; hence it must 
be one of the three chief planes ; hence the diameter of the cen- 
tral circle must be one of the axes. This can only be the mean 
one, 26, in case of the ellipsoid; for any circle of radius a 



270 CO-OEDINATE GEOINIETEY. 

resp. c lies wholty without resp. within the ellipsoid. Similar 

reasoning holds for the other surfaces. But there are other 

imaginary cyclic planes^ as may thus be shown analytically : 

x^ if- z- o(? y- z^ 

Be -5 -f -7, + — = 1 an ellipsoid, and —,+ --„ 4- — = 1 a 

concentric sphere ; then 



is a cone through 0, being homogeneous of second degree, and 
through the intersection of sphere and ellipsoid, being satisfied 
whenever their Eqs. are. When, and only luhen, this cone 
breaks up into two planes, the intersection of ellipsoid and 
sphere is a plane curve ; i.e., is a circle. Tliis is the case only 
when the Eq. is resoluble into two linear factors ; and this is 
the case onh' when the determinant A vanishes, or when one 
coefficient (one element in the diagonal of A, the others being 
0) vanishes ; and this is so onl}^ when ?*^ = a^, or 6^, or c^. 
For i^=^a' or ?'^=:c^, the factors, i.e., the planes, are 
imaginary; for i"^ =i}/ they are rea^. The student can easily 
apply the reasoning to the other surfaces. 

55. We have seen that two RLs. lie on the hyperbolic para- 
boloid: the intersection of that surface and the Xy-plane. 
But the general proposition holds : 

All surfaces of second degree are ruled. On each lies an oc 
of RLs. This is clear at once on referring to the condition that 
a E,L. lie on a surface (Art. 17) : on combining the Eqs. of E-L. 
and surface, the resultant Eq. in a single Cd. must vanish iden- 
tically. This Eq., being of second degree, vanishes thus when 
its three coefficients each reduce to ; and these three Eqs. of 
condition can be satisfied by the four parameters of a RL. in an 
00 of ways. 

56. Let us apply this argument to the simple hyperboloid : 

o? b^ & 



EIGHT LINES ON THE QUADKIC 271 



Be y=tz-\-i\ x = sz-\-u 

the RL. On substitution results 



This vanishes identically, is satisfied for every 2;, when 

^^ 4- ^^ _ -'■ — n su tv _n^ '^^^ I "^^ _ 1 
o? W (^ ' d? h^ ^ o? IP 

Hence result readily the real values 
s = ± av : 6c, t = ^ bu : ac. 

The third Eq. of condition says that every such RL. meets the 
ellipse of the gorge, as was to be foreseen. The double sign 
shows that through every point of this ellipse go two RLs. ; 
there lies on the surface a double system of RLs. Two RLs., 
one of each sj^stem, are 

av , hi 
x = — z-\-u, y = z + v, 

be ac 

, av' , , bu' , , 

and x = z-\-u', y= — z-\-v'. 

be ac 

The condition that these two RLs. meet is (Art. 9) 

, , av , av' fbu , bu'\ 
u — u' :v — v' = 1 : — 1 , 

be be \ac ac) 

a condition always fulfilled, since 

Hence every RL. of each system meets every RL. of the other. 

av bu 

Changing the signs of — and — , we see that the con- 

bc ac 

dition is fulfilled only when {u — u'Yb"^ ■= — a^ {v — v'Y \ i.e., 

never. Hence no RL. of either system meets any RL. of the 

same system. Hence through everj- point of the surface there 

pass two^ and only two, RLs. on it. ThQ plane of these RLs. is 



272 COOEDINATE GEOMETEY. 

clearly the plane tangent at their intersection. For it can meet 
the surface only on these RLs., wJiich form its couic of inter- 
section ; hence all RLs. in it throogh the intersection of the 
pair meet the surface only at that point. As the point of tan- 
gence glides along either of the RLs., the tangent plane turns 
about the RL., cutting the surface. 

The RLs. of either sj^stem are called elements (or generators) 
of the surface. Since no two elements meet, the surface is not 
torse or developable^ but skew or a scroll. This is easy to see, 
thus: Be 1, 2, 3, 4, ... consecutive elements. If 1 and 2 
meet, 2 and 3 meet, 3 and 4 meet, etc., then we may turn the 
strip between 1 and 2, which is plane, infinitesimally, about 2 
till it falls into the plane of the strip 2 3 ; then turn the sum of 
the strips 1 2 and 2 3 about 3 into the plane of the strip 3 4, 
and so on. Thus, and thus only, could the surface be turned 
off, unwrapped, into a plane surface. Now, since 1 and 2, 
2 and 3, etc., do not meet, this can not be done. A more 
rigorous proof would not be in place here. 

57. To find the RLs. on an ellipsoid^ in the values of s and t^ 
put c^ for — c^, or ic for c; the values then fall out imaginary: 
there are no real RLs. on the ellipsoid. 

On putting a^, hi for a, 6, the values of s and t again fall out 
imaginary : no real RLs. lie on the double hyperboloid. 

58. Proceeding with the hyperbolic paraboloid — — ^ = 42 

exactly as with the simple hyperboloid, we find for s and t the 
real values : s = a : zi, t= ± -Vab : u. Hence there lies on it 
a double sj'stem of real RLs. Every RL. of either system cuts 
every RL. of the other. No RL. of either system cuts a RL. 
of the same system. Through every point of the surface pass 
a pair of RLs., fixing the tangent plane at the point, which 
plane cuts the surface. The surface is not developable. 

Eliminating 2;, the student will find the X3^-projection of an 
element to be 

y = ±A{x-2u)', 

\a 



COi^FOCAL QUADRICS. 273 

whence projections of all elements, on XF, are I| to one of the 
pah' of RLs. 2/ = ± V^ • ^^ ' ^i ^^ which XY cuts the surface ; 
i.e., all elements are || to one of the planes fixed by Z and this 
pair. These planes contain the asymptotes of the generating 
hyperbola. Hence all elements of a JiyperboUc paraboloid are 
II to an asymptotic plane. (See Figs, at end.) 

59. The foci of the chief sections of a quadric are called foci 
of the quadric. In an ellipsoid with half-axes a, 6, c, the sec- 
tion {ah) is an £ with two foci F^ F' on the axis 2 a, distant 
Va^ — b'^ from the centre ; the section (be) is an £ with two foci 
G^ G' on the axis 2&, distant ■\^b- — c^ from the centre; the 
section (ac) is an £ with two foci H, S' on the axis 2 a, distant 
Va^ — c^ from the centre. Thus, on the greatest axis lie four 
foci, on the mean axis lie two, on the least lie none. 

If a quadric with half-axes a\ b\ c' be confocal with this 
base-ellipsoid, the relations hold : 

a'2-6'2 = a2-62, b" - c'^^b"" -(f, a"" - c^^ = a^ - c^ ; 
whence 

a'2 -a' = b'- -b' = c'- - c^ = (say) X ; 

i.e., ^+4+4 = 1, and _^1- + JL- + ^ = 1, 
' a'^ b' c' ' a^ + A 52 + X c^ + A 

are confocal for all values of X. 

To trace the S3'stem : for A = + 00 the surface is a sphere 
with radius 00 ; as A sinks toward — c^, the surface (an ellip- 
soid) shrinks, and for A = — c^ flattens to the inner doubly- 
laid surface of the so-called focal E in the section (ab) , whose 
half-axes are Va^ — c-, V6^ — c^, its foci F., F\ and its vertices 
H^ iT ', G^, & ; as A sinks from — c^ towards — 6", the outer 
doubly-laid surface (tliought as a simple hyperboloid) spreads 
out into a simple hyperboloid, which, as A nears —6^, flattens 
into the so-called focal H in the section {ac) with foci H^ H' 
and vertices F, F' ; as A sinks from — b^, the surface becomes a 
double hyperboloid, which, as A nears — a^, flattens down to the 



274 CO-OIIDINATE GEOMETRY. 

section (5c); as A sinks from — a^ towards — oo, the surface 
becomes and remains an imaginary ellipsoid. 

60. For any triplet (a?', ?/', 2;') the Eq. of the confocal yields 
three values of A : Aj, A2, A3 ; by reasoning quite like that in 
Plane Geometry (Art. 148), it is shown that these roots lie 
between + 00 and — c", — c^ and — 6^ — 6^ and — a^, resp. ; 
i.e., through every poi7it of sjxice pass three, and only three, 
confocals: an ellipsoid, a simple hyperboloid, and a double 
hyperboloid. 

The three A's are called elliptic Cds. of the point (a?', y', z') . 
Substituting them in the Eq. of the confocal, in turn, and solv- 
ing the three Eqs. as to x', y',z\ we get 



,2; 



ic' = -v/ J a- -h Aj • a^ + A2 • a^ + A3 J : ->/ J a^ — 6^ • a^ — c^ 

and two like expressions for y\ z' got by permuting a, &, c. 

If we divide this x' by a^ + A^, we get the coefficient of x in 
the Eq. of the plane tangent at (a;', y', z') to the first confocal ; 
dividing it by a^+A2, we get the corresponding coefficient in the 
Eq. of the plane tangent to the second confocal ; the product 
of these two coefficients is (cr + A3) : (a^ — 6^) (a^ — c^). The 
products of the coefficients of y and of z in the two Eqs. are 
got by simply permuting a,b,c. The sum of these three prod- 
ucts is 0. This means, by Art 16, that' the two planes are _L. 
Like holds, of course, for the second and third confocals, and 
for the third and first. Hence three confocals through a point 
are mutually _L. 

Cubature of the Quadric. 

61. The part of a surface intercepted between two \\ planes is 
called a zone. The space bounded b}^ the planes and the zone 
we may call a segment of the surface (meaning a segment of the 
space fixed by the surface) . 

Suppose an equiaxial H, its conj. H\ and their common 
asymptotes turned about the conjugate axis. There will be 
generated by H, a simple hyperboloid of rcA^olution ; by H\ a 



CUBATUSE OF THE QUADEIC. 275 

double hyperholoid of revolution ; by the asymptotes, a cone of 
revolution : the Eqs. are 

a? + y^ — z^ = o?^ y? -\-if — z^ =^ —o?^ x^ ■\-y'^ — z^=0. 

Sections of the surfaces 1. to Z are circles, and their areas, 
they being distant z from XY, are 

TT (z^ + a^) 5 TT (z^ — a^) , ttz^. 

Hence it is clear that the circle of the cone is the arithmetic 
mean between the cu'cles of the hyperholoids ; it differs from 
each of these by a ring whose area is 7ra^, the area of the circle 
of the gorge; these differ from each other hy double this area, 
by 2 ira^. It is to note that the circle of the double hyperboloid 
is imaginary^ and so does not really come into consideration, for 
z<ia. 

Accordingly, to find the volume of any hyperboloidcd seg- 
ment^ it suffices to find the volume of the corresponding cone- 
segment and then add resp. sttbtract the volume of the corre- 
sponding ring-segment in case of the single resp. double 
hyperboloid. The cone-segment is itself the difference of two 
cones whose altitudes are (say) Zi, z^-, and bases ttz^^ ttz^ ; 

hence the volume is - (z^^ — z-^) ; the constant area of a section 

of the ring-space is Trcr, and the altitude is Z2 — Zi\ hence the 
volume is ira^ {z2 — Zi) . 

If the H be not equiaxial, but have axes 2 a, 2 c, then to any 
altitude z will correspond in the cone a circle of radius not z but 

-z, the surfaces then being 
c 

2 2 

, n Ct I 1 a n 9 

ar + / -.z- = a-, x--{- y' — — 'Z^ = -a\ 

cr cr 

2 

9,0 a 9 r\ 

Hence it is enough to change z into az : c. From these la?t 
surfaces the most general, viz., 



276 CO-OHDINATE GEOMETRY. 



0? ip- C^ G^ W (? 






^2 ' 7,a ^2 ' 



are got by shortening every y in the ratio 5 ; a ; hence it suf- 
fices to multiply the preceding results by this ratio. 

62. The ellipsoid (a, &, c) is got froni the sphere (a, a, a) by 
shortening every y in the ratio 6 : a, and every z in the ratio 
Q,\a\ hence the whole volunie of the ellipsoid is got from that 
of the sphere by shrinking it in the ratio hc:o? \ and the same 
ratio holds between volumes of corresponding parts of ellipsoid 
and sphere. The volume of the sphere is l-n-a'^, hence that of 
ellipsoid is |ira6c. 

On three axes, 2 a, 2 6, 2 c, construct an ellipsoid and the two 
Tiyperholoids ; also construct a cylinder tangent to the simple 
hyperboloid along the ellipse of the gorge, its bases tangent to 
the double hyperboloid at the latter' s vertices. Let us compare 
the volumes E, (7, /iT, II of the ellipsoid, cylinder, cone-seg- 
ment, hyperboloid-segment, the bases of the two latter being 
the bases of the cylinder. The volume E is ^Trabc ; C is 2 c • irab 
or 2 -n-abc ; K is ^ of (7, or is ^-rrabc ; // is K plus 2 c • -n-ab, or H 
is ^irabc ; hence 

K'.EtCiII=lt2:nU. 

63. To find the volume V of a segment of the elliptic parabo- 

loid }- ^ = 42, first take the vertex for one base and the 

a 

section of the plane z = z for the other ; cut- this cap-shaped 

segment into n thin slices by planes H to the base ; let the alti- 

z 
tude of each slice be - ; it will have two bases, each an £, a 

n 

greater and a less; the volume of each slice will be less than the 
altitude by the greater base and greater than the altitude by the 
less base ; hence the whole volume will be less than the common 

altitude - by the sum of the greater bases and greater than that 



METHOD OF SLICES. 277 

altitude b}' the sum of the smaller bases ; or, common factors 
set out, 

F>— -^TrV^iO + l+SH \-n-l], 

or V< \ 1 + - 

2 {, n 

-rr.^ 4 1T-\I ah 'Z'Z { ^ 1 

2 \ ~n 

1 

As n nears oo, - nears 0, and there results 
n 

V=2'trVab,z,z, 

Now 4 7r-\/ab ' z is the base of the segment, z its altitude ; 
hence 4 tt Va& • 2; • ;<; is the volume of the circumscribing cylinder ; 
hence the volume of a cap-segment of an elliptic p>araboloicl is 
half that of the circumscribed cylinder. 

The volume of any segment is the difference of two cap- 
segments. 

64. To find the volume v of a segment of an hyperbolic parab- 

oloid —-— ^^= iz, suppose it bounded by the surface, the 
or h ^ 

Xy-plane, the y^T- plane, and a plane x = x^ II to YZ. The 

section of this last plane with the surface is a parabola ; the 

chord (in Xy-plane) of the segment of this P is 2iX^b%a^ the 

altitude of the segment is i;c^;4a ; hence the area is 

y/ab,QO^ 



3 . a- 



Cut the solid segment into n thin slices ; then, reasoning exactly 
as before, we get 



3 • anr 



278 CO-OEDINATE GEOMETKY. 



V«6 • X" 



v> 1 : 1 103+134,234-...+^ 



Now 13 + 2^+. ..+71^=: ^^^^ + ^^ , 

4 
as we know from Algebm ; hence 

l'2cr \ vj 

^ V^Wa_iv 

V2a? \ n 

As n nears oo , - nears 0, and there results 
n 



Now Va&-i«^:3 cr is the base of the segment, x its altitude ; 
hence -y/ab-x^-.Sa' is the volume of the circumscribed cylinder; 
hence the volume of such a segment of an hyperbolic paraboloid 
is one-fourth that of the circumscribed cylinder. 

The student may confirm the results as to the ellipsoid and 
the hyperboloids by this method of slices. 

The segments thus far treated have been rights i.e., _L to an 
axis of the surface ; but like reasoning applies to oblique seg- 
ments, on observing that the intercept between the bases on the 
conjugate diameter is not the altitude of the segment but a 
multiple of it. 

Varieties of Quadrics. 

65. If a quadric be given by its Eq. in the general form, it is 
of course possible to determine what kind of a quadric it is by 
reducing its Eq. to the simplest form ; but this is tedious. It 
is possible to establish certain simple tests, however, b}' some 
such reasoning as this : 

By Art. 30 the surface is centric or non-centric^ according 
as D>0 or D = 0. If D^O, the centric Eq. is 

koi? + 2 hxy -\-jy^+ 2gzx + 2fyz + iz^-^ — = 0. 



VARIETIES OF QUADBICS. 279 

If now a RL. through the centre, y — tz, x = uz, meet 
the surface in finity for all values of t and u, the surface is 
closed or ellipsoidal; otherwise, it is hype^^holoidal. The stu- 
dent can show that the first holds when both D > and G or 
fij _ 7^2 ^ ; the surface is then a real ellipsoid^ an imaginary 
cone with one real point (the centre), or an imaginary ellipsoid, 
according as A<0, A = 0, or A>0 (/j being taken + 
alwa3's) . 

If D and C be not both >0, the surface is hyperboloidal. 
We now test whether the ELs. on it be real or imaginary by the 
former method ; the result is, the surface is a simple hyperbo- 
loid, an elliptic co7ie, or a double hyperholoid, according as 

A<0, A = 0, A<0. 

In case D = 0, the surface is non-centric or paraholoidal. 
Putting z — 0, we find the section of the XF-plane is an E or 
an //, i.e., the surface is an elliptic or an hyperbolic pctmSoZoic?, 
according as C>0 or C<0. If (7=0, the section is a P, 
and this test fails. In that case, test with ik — g'^ in the same 
way. If both kj — W and ik — g"^ vanish, then must also ji —p 
vanish ; all sections are /''s, and the surface is a parabolic 
cylinder. 

Lastly, in case one or more of the numerators i, Jf, N of the 
Cds. of the centre (i : i), M'. D, N-. D) , as well as the common 
denominator D, vanish, the centre becomes indeterminate, the 
surface has an oo of centres. The surface is then a cylinder. 
In case the three Eqs. of planes Q^ = 0, Q^ = 0, Q^ = 
which determine the centre reduces to two only, their line of in- 
tersection is the line of centres, every point on it is a centre of 
the surface. The cylinder is elliptic, hyperbolic, or breaks up 
into 2^ pair of planes, according as one of its plane sections is an 
£, an H, or a pair o/ intersecting RLs. In case the three Eqs. 
reduce to one, each represents iho, plane of centres, every point 
on it is a centre of the surface. The surface itself consists of 
two II planes, midwa}^ between which lies the \\ plane of centres. 
In case the Eq. of the surface is a perfect square, the surface 
consists of tioo planes fcdlen together in the plane of centres. 



280 



CO-OE,DIKATE GEOMETJSY. 




^ip;. 3 



Ellipsoid. 
OA=a, OB = b, OC=c. 














-.^ 




^ J--' 


^ 


-^^^ 


^^ / 


/ 


\ 


^*^^>~^^ 


y/^ / 


'' 


\ 


^N^^S. 






/ 
/ 
1 
1 


^\\ 


V ^ ''^ 





1 
\ 
\ 
\ 


il) 


x^ «^ 




/ 


^"^^ r^^ 


\^^ \ 


\ 


/ 


/^•^^ 




V 


/ 


^^^^ 


-— ::i_ 


1 ^- 




^^ 



A 



B 



Pkojections of Ellipsoid 
on xz and xy. 




Simple Hyperboloid. 

AB is Ellipse of the Gorge. 
EOD is the Asymptotic Cone. 



Ifis. 4r 




OD gives direction of cyclic sections. 

Cs, are cyclic points. 

C'C is locus of centres of cyclic sections. MP^ and NQ^ are of 2d system. 



ELs. ON THE S. H. 

MP^ and NQ^ are of 1st system, 



Note. For Figures on pp. 280 and 281 thanks are due Fort and Schloemilch's 

A».alytische Geometrie. 



DIAGRAMS. 



281 




Hyperbolic Paraboloid. 

OG and OB are Parabolas. 
OA and OB are Asymptotic directions 
for tiie Hyperbolas. 



Double Htperboloid. 
EOD is the Asymptotic Cone. 




Elliptic Paraboloid. 

FA = 2a and GB = 2b are 

half -parameters. 




ELs. on the H. p. 



^«ss of 
§ioglon. 



Peirce's Three and Four Place Tables of Loga- 

rithmic and Trigonometric Functions. By James Mills Peirce, 
University Professor of Mathematics in Harvard University. Quarto. 
Cloth. Mailing Price, 45 cts. ; Introduction, 40 cts. 

Four-place tables require, in the long run, only half as much time 
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Elements of the Differential Calculus. 

With Numerous Examples and Applications. Designed for Use as a 
College Text-Book. By W. E. Byerly, Professor of Mathematics, 
Harvard University. 8vo. 273 pages. Mailing Price, $2.15 ; Intro- 
duction, ^2.00. 

This book embodies the results of the author's experience in 
teaching the Calculus at Cornell and Harvard Universities, and is 
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up the interest of the student by bringing in throughout the whole 
book, and not merely at the end, numerous applications to practical 
problems in geometry and mechanics. 



James Mills Peirce, Prof, of 
Math., Harvard Univ. (From the Har- 
vard Register^ : In mathematics, as in 
other branches of study, the need is 
now very much felt of teaching which 



is general without being superficial; 
limited to leading topics, and yet with- 
in its limits; thorough, accurate, and 
practical ; adapted to the communica- 
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as knowledge, but free from details 
which are important only to the spe- 
cialist. Professor Byerly's Calculus 
appears to be designed to meet this 
want. . . . Such a plan leaves much 
room for the exercise of individual 
judgment ; and differences of opinion 
will undoubtedly exist in regard to one 
and another point of this book. But 
all teachers will agree that in selection, 
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spirit in the learner. . . . The book 
contains, perhaps, all of the integral 
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relations of abstract thought, and is 
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idea, ought to be familiar. One who 
aspires to technical learning must sup- 
plement his mastery of the elements 
by. the study of the comprehensive 
theoretical treatises. . . . But he who is 
thoroughly acquainted with the book 
before us has made a long stride into 
a sound and practical knowledge of 
the subject of the calculus. He has 
begun to be a real analyst. 

H. A, Ne-wton, Prof, of Math, in 
Yale Coll., New Haven : I have looked 
it through with care, and find the sub- 
ject very clearly and logically devel- 
oped. I am strongly inclined to use it 
in my class next year. 

S. Hart, recent Prof, of Math, in 
Trinity Coll., Conn. : The student can 
hardly fail, I think, to get from the book 
an exact, and, at the same time, a satis- 
factory explanation of the principles on 
which the Calculus is based ; and the 
introduction of the simpler methods of 



integration, as they are needed, enables 
applications of those principles to be 
introduced in such a way as to be both 
interesting and instructive. 

Charles Kraus, Techniker,Pard- 
icbitz, Bohemia, Austria : Indem ich 
den Empfang Ihres Buches dankend 
bestaetige muss ich Ihnen, hoch geehr- 
ter Herr gestehen, dass mich dasselbe 
sehr erfreut hat, da es sich durch 
grosse Reichhaltigkeit,besonders klare 
Schreibweise und vorzuegliche Behand- 
lung des Stoffes auszeichnet, und er- 
weist sich dieses Werk als eine bedeut- 
ende Bereicherung der mathematischen 
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De Volson "Wood, Prof of 
Math., Stevens' Inst., Hoboken, N.J. : 
To say, as I do, that it is a first-class 
work, is probably repeating what many 
have already said for it. I admire the 
rigid logical character of the work, 
and am gratified to see that so able a 
writer has shown explicitly the relation 
between Derivatives, Infinitesimals, and 
Differentials. The method of Limits 
is the true one on which to found the 
science of the calculus. The work is 
not only comprehensive, but no vague- 
ness is allowed in regard to definitions 
or fundamental principles. 

Del Kemper, Prof of Math., 
Hampden Sidney Coll., Va. : My high 
estimate of it has been amply vindi- 
cated by its use in the class-room. 

R. H. Graves, Prof of Math., 
Univ. of North Carolina : I have al- 
ready decided to use it with my next 
class ; it suits my purpose better than 
any other book on the same subject 
with which I am acquainted. 

Edw. Brooks, Author of a Series 
of Math. : Its statements are clear and 
scholarly, and its methods thoroughly 
analytic and in the spirit of the latest 
mathematical thought. 



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Elements of the Integral Calculus. 

By W. E. Byerly, Professor of Mathematics in Harvard University. 
Svo. 204 pages. Mailing Price, ^2.15; Introduction, 32.00. 

This volume is a sequel to the author's treatise on the Differential 
Calculus (see page 134), and, like that, is written as a text-book. 
The last chapter, however, — a Key to the Solution of Differential 
Equations, — may prove of service to working mathematicians. 



H. A, Newton, Prof, of Math., 
Yale Coll. : We shall use it in my 
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Mathematical Visitor : The 
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culus that we have seen which devotes 
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A Short Table of Integrals. 



To accompany BYERLY'S INTEGRAL CALCULUS. By B. O. 
Peirce, Jr., Instructor in Mathematics, Harvard University. i6 pages. 
Mailing Price, lo cts. To be bound with future editions of the Calculus. 



Elements of Quaternions. 



By A. S. Hardy, Ph.D., Professor of Mathematics, Dartmouth College. 
Crown, 8vo. Cloth. 240 pages. Mailing Price, $2.15; Introduction, 
^2.00. 

The chief aim has been to meet the wants of beginners in the 
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The Introduction to Qitaterjiions by Kelland contains many exer- 
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instruction common in this country. 

PKESS NOTICES. 



Westminster Review : It is a 

remarkably clear exposition of the sub- 
ject. ■ 

The Daily Review, Edinburgh, 
Scotland : This is an admirable text- 
book. Prof. Hardy has ably supplied 
a felt want. The definitions are models 
of conciseness and perspicuity. 



The Nation : For those who have 
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seems to us superior both to the work 
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Profs. Tait and Kelland. 

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certainly need search no longer for 
Quaternions made plain. 

Van Nostrand Eng-ineering 
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a glance at the opening chapter of 
Prof. Hardy's work will enforce the 
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is quite opportune. The subject must 



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London Schoolmaster : It is in 

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Calculus, with its concise notation, is a 
most powerful instrument for mathe- 
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Boston Transcript : A text-book 

of unquestioned excellence, and one 
peculiarly fitted for use in American 
schools and colleges. 

The Western, St. Louis: This 
work exhibits the scope and power of 
the new analysis in a very clear and 
concise form . . . illustrates very finely 
the important fact that a few simple 
principles underlie the whole body of 
mathematical truth. 



FROM COLLEGE PROFESSORS. 



James Mills Peirce, Prof, of 
Math., Harvard Coll. : I am much 
pleased with it. It seems to me to 
supply in a very satisfactory manner 
the need which has long existed of a 
clear, concise, well-arranged, and logi- 
cally-developed introduction to this 
branch of Mathematics. I think Prof. 
Hardy has shown excellent judgment 
in his methods of treatment, and also 
in limiting himself to the exposition 
and illustration of the fundamental 
principles of his subject. It is, as it 



ought to be, simply a preparation for 
the study of the writings of Hamilton 
and Tait, I hope the publication of 
this attractive treatise will increase the 
attention paid in our colleges to the 
profound, powerful, and fascinating cal- 
culus of which it treats, 

Charles A. Young, Prof, of 
Astrononty, Princeton Coll. : I find it 
by far the most clear and intelligible 
statement of the matter I have yet 
seen. 



Elements of ike Differential and Integral Calculus. 

With Examples and Applications. By J. M. Taylor, Professor of 
Mathematics in Madison University. 8vo. Cloth. 249 pp. Mailing 
price, $1.95; Introduction price, ^1.80. 

The aim of this treatise is to present simply and concisely the 
fundamental problems of the Calculus, their solution, and more 
common applications. Its axiomatic datum is that the change of a 
variable, when not uniform, may be conceived as becoming uniform 
at any value of the variable. 

It employs the conception of rates, which affords finite differen- 
tials, and also the simplest and most natural view of the problem of 
the Differential Calculus. This problem of finding the relative 
rates of change of related variables is afterwards reduced to that of 
finding the limit of the ratio of their simultaneous increments ; and, 
in a final chapter, the latter problem is solved by the principles of 
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Judging from the author's experience in teaching the subject, it 
is believed that this elementary treatise so sets forth and illustrates 
the highly practical nature ef the Calculus, as to awaken a lively 
interest in many readers to whom a more abstract method of treat- 
ment would be distasteful. 



Oren Root, Jr., Prof, of Math., 
Hamilton Coll., N.Y.: In reading the 
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densation. It seems to me most admir- 
ably suited for use in college classes. 
I prove my regard by adopting this as 
our text-book on the calculus. 



C. M. Charrappin, S.J., St. 

Louis Univ. : I have given the book a 
thorough examination, and I am satis- 
fied that it is the best work on the sub- 
ject I have seen. I mean the best 
work for what it was intended, — a text- 
book. I would like very much 10 in- 
troduce it in the University. 
(yan. 12, 1885.) 



